Set families t2From Wikipedia, the free encyclopedia

Contents

1 Abstract simplicial complex 11.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Almost disjoint sets 52.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Other meanings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Antimatroid 73.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Paths and basic words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Join-distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.6 Supersolvable antimatroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.7 Join operation and convex dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.8 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Block design 144.1 Denition of a BIBD (or 2-design) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Symmetric BIBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 Projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 Biplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 Hadamard 2-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3 Resolvable 2-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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4.4 Generalization: t-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Derived and extendable t-designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.5 Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.6 Partially balanced designs (PBIBDs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.6.3 Two associate class PBIBDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Carathodorys theorem (convex hull) 225.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Clique complex 256.1 Independence complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Flag complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.3 Conformal hypergraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Examples and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7 Combinatorial design 287.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.2 Fundamental combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.3 A wide assortment of other combinatorial designs . . . . . . . . . . . . . . . . . . . . . . . . . . 307.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 Content (measure theory) 378.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.2 Integration of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.3 Duals of spaces of bounded functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4 Construction of a measure from a content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.4.1 Denition on open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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8.4.2 Denition on all sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.4.3 Construction of a measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

8.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

9 Dedekind number 409.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.3 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.4 Summation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.5 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

10 Delta set 4410.1 Denition and related data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.2 Related functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

11 Delta-ring 4811.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4811.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

12 Dendroidal set 4912.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4912.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

13 Discrete dierential geometry 5013.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5013.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14 Disjoint sets 5114.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.3 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.4 Disjoint unions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5314.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

15 DoldKan correspondence 5515.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5515.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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15.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16 Dynkin system 5616.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5716.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

17 ErdsKoRado theorem 5817.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5817.2 Families of maximum size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.3 Maximal intersecting families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

18 Family of sets 6118.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.2 Special types of set family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6218.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

19 Field of sets 6319.1 Fields of sets in the representation theory of Boolean algebras . . . . . . . . . . . . . . . . . . . . 63

19.1.1 Stone representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.1.2 Separative and compact elds of sets: towards Stone duality . . . . . . . . . . . . . . . . . 63

19.2 Fields of sets with additional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2.1 Sigma algebras and measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2.2 Topological elds of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6419.2.3 Preorder elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6519.2.4 Complex algebras and elds of sets on relational structures . . . . . . . . . . . . . . . . . . 65

19.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6619.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

20 Finite character 6720.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

21 Finite intersection property 6821.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6821.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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21.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6921.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

22 Fishers inequality 7022.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.2 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7122.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

23 Generalized quadrangle 7223.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7323.5 Generalized quadrangles with lines of size 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7423.6 Classical generalized quadrangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7423.7 Non-classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.8 Restrictions on parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7523.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

24 Greedoid 7624.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7624.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7724.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

25 Helly family 7925.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7925.3 Helly dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.4 The Helly property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8025.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

26 Hellys theorem 8226.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8226.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8326.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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26.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

27 Incidence structure 8527.1 Formal denition and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8527.3 Dual structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.4 Other terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

27.4.1 Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8627.4.2 Block designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

27.5 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.5.1 Incidence matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.5.2 Pictorial representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8827.5.3 Incidence graph (Levi graph) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

27.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9127.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9227.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

28 Kan bration 9328.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9328.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

29 Kirkmans schoolgirl problem 9729.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9729.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.3 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.4 Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

30 KruskalKatona theorem 10230.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

30.1.1 Statement for simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10230.1.2 Statement for uniform hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

30.2 Ingredients of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10230.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10330.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

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31 Levi graph 10431.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10431.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10531.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

32 Matroid 10632.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

32.1.1 Independent sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10632.1.2 Bases and circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10632.1.3 Rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.1.4 Closure operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10732.1.5 Flats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.1.6 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

32.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.2.1 Uniform matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.2.2 Matroids from linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10932.2.3 Matroids from graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11132.2.4 Matroids from eld extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

32.3 Basic constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.3.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.3.2 Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11232.3.3 Sums and unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

32.4 Additional terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.5 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

32.5.1 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11332.5.2 Matroid partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.5.3 Matroid intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.5.4 Matroid software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

32.6 Polynomial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.6.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11432.6.2 Tutte polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

32.7 Innite matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.8 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11632.9 Researchers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11732.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11732.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11732.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11932.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

33 Maximum coverage problem 12133.1 ILP formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12133.2 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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33.3 Known extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12233.4 Weighted version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12233.5 Budgeted maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12233.6 Generalized maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

33.6.1 Generalized maximum coverage algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.7 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12333.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

34 Monotone class theorem 12434.1 Denition of a monotone class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12434.2 Monotone class theorem for sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

34.2.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12434.3 Monotone class theorem for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

34.3.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12434.3.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

34.4 Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12534.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

35 Near polygon 12635.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12735.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12735.3 Regular near polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12735.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12835.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12835.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

36 Nerve (category theory) 12936.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12936.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12936.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

36.3.1 Most spaces are classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13136.3.2 The nerve of an open covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13136.3.3 A moduli example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

36.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

37 Nerve of a covering 13337.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13337.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

38 Noncrossing partition 13438.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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38.2 Lattice structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.3 Role in free probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13438.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

39 Partition of a set 13839.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.3 Partitions and equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13939.4 Renement of partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.5 Noncrossing partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.6 Counting partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14039.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14139.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14139.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

40 Partition regularity 14640.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14640.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

41 Pi system 14841.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14841.2 Relationship to -Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

41.2.1 The - Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14941.3 -Systems in Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

41.3.1 Equality in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15041.3.2 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

41.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15141.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15141.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

42 Polar space 15242.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15242.2 Classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15242.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

43 Pro-simplicial set 15343.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

44 Property B 15444.1 Values of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15444.2 Asymptotics of m(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15444.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

45 Radons theorem 156

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45.1 Proof and construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15645.2 Topological Radon theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15745.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15745.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15745.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15845.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

46 Ring of sets 16046.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16046.2 Related structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16146.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16146.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

47 SauerShelah lemma 16247.1 Denitions and statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16347.2 The number of shattered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16347.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16347.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16347.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

48 Segal space 16548.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16548.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

49 Set cover problem 16649.1 Integer linear program formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16649.2 Hitting set formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16649.3 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16649.4 Low-frequency systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16749.5 Inapproximability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16749.6 Related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16749.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16849.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16849.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

50 ShapleyFolkman lemma 17050.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17150.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

50.2.1 Real vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17150.2.2 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17250.2.3 Convex hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17350.2.4 Minkowski addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17350.2.5 Convex hulls of Minkowski sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

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50.3 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17550.3.1 Lemma of Shapley and Folkman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17550.3.2 ShapleyFolkman theorem and Starrs corollary . . . . . . . . . . . . . . . . . . . . . . . 17750.3.3 Proofs and computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

50.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17850.4.1 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17950.4.2 Mathematical optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18350.4.3 Probability and measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

50.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18550.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18950.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

51 Sigma-algebra 19151.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

51.1.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19151.1.2 Limits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19251.1.3 Sub -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

51.2 Denition and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.2.2 Dynkins - theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.2.3 Combining -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19351.2.4 -algebras for subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19451.2.5 Relation to -ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19451.2.6 Typographic note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

51.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.3.1 Simple set-based examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.3.2 Stopping time sigma-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

51.4 -algebras generated by families of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.4.1 -algebra generated by an arbitrary family . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.4.2 -algebra generated by a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19551.4.3 Borel and Lebesgue -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19651.4.4 Product -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19651.4.5 -algebra generated by cylinder sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19651.4.6 -algebra generated by random variable or vector . . . . . . . . . . . . . . . . . . . . . . 19751.4.7 -algebra generated by a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . 197

51.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19751.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19851.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

52 Sigma-ideal 19952.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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53 Sigma-ring 20053.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.3 Similar concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.4 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20053.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20153.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

54 Simplex category 20254.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20254.2 Augmented simplex category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20254.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20254.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20354.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

55 Simplicial approximation theorem 20455.1 Formal statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20455.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

56 Simplicial complex 20556.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20656.2 Closure, star, and link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20656.3 Algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20656.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20756.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20756.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20756.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

57 Simplicial group 20957.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20957.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

58 Simplicial homotopy 21058.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21058.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

59 Simplicial manifold 21159.1 A manifold made out of simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21159.2 A simplicial object built from manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

60 Simplicial presheaf 21260.1 Homotopy sheaves of a simplicial presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21260.2 Model structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21260.3 Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

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60.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21360.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21360.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21360.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21360.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

61 Simplicial set 21461.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21461.2 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21461.3 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21561.4 Face and degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21561.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21561.6 The standard n-simplex and the category of simplices . . . . . . . . . . . . . . . . . . . . . . . . . 21661.7 Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21661.8 Singular set for a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21761.9 Homotopy theory of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21761.10Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21761.11History and uses of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21861.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21861.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21861.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

62 Sperners theorem 22062.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22062.2 Partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22062.3 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22062.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

62.4.1 No long chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.4.2 p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.4.3 No long chains in p-compositions of a set . . . . . . . . . . . . . . . . . . . . . . . . . . . 22162.4.4 Projective geometry analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22262.4.5 No long chains in p-compositions of a projective space . . . . . . . . . . . . . . . . . . . . 222

62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22262.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

63 Steiner system 22463.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

63.1.1 Finite projective planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22563.1.2 Finite ane planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

63.2 Classical Steiner systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22563.2.1 Steiner triple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22563.2.2 Steiner quadruple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

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63.2.3 Steiner quintuple systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22563.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.5 Mathieu groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22663.6 The Steiner system S(5, 6, 12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

63.6.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22763.7 The Steiner system S(5, 8, 24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

63.7.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22863.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22863.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22963.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22963.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

64 Symmetric spectrum 23164.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

65 TeichmllerTukey lemma 23265.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23265.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23265.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23265.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

66 Tverbergs theorem 23366.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23466.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23466.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

67 Two-graph 23567.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23567.2 Switching and graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23567.3 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23767.4 Equiangular lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23767.5 Strongly regular graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23767.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23767.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

68 Ultralter 23968.1 Formal denition for ultralter on a set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23968.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24068.3 Generalization to partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24068.4 Special case: Boolean algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24068.5 Types and existence of ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24068.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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68.7 Ordering on ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24268.8 Ultralters on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24268.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24268.10Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24268.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

69 Union-closed sets conjecture 24469.1 Equivalent forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24469.2 Families known to satisfy the conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24469.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24569.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24569.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24569.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

70 Universal set 24670.1 Reasons for nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

70.1.1 Russells paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24670.1.2 Cantors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

70.2 Theories of universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24670.2.1 Restricted comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24770.2.2 Universal objects that are not sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

70.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24770.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24770.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

71 Universe (mathematics) 24971.1 In a specic context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24971.2 In ordinary mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24971.3 In set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25071.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25171.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25271.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25271.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25271.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

72 VietorisRips complex 25372.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25472.2 Relation to ech complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25472.3 Relation to unit disk graphs and clique complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 25472.4 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25472.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25472.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25572.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

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73 -groupoid 25673.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25673.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25673.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 257

73.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25773.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26173.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Chapter 1

Abstract simplicial complex

In mathematics, an abstract simplicial complex is a purely combinatorial description of the geometric notion of asimplicial complex, consisting of a family of non-empty nite sets closed under the operation of taking non-emptysubsets.[1] In the context of matroids and greedoids, abstract simplicial complexes are also called independencesystems.[2]

1.1 DenitionsA family of non-empty nite subsets of a universal set S is an abstract simplicial complex if, for every set X in, and every non-empty subset Y X, Y also belongs to .The nite sets that belong to are called faces of the complex, and a face Y is said to belong to another face X if Y X, so the denition of an abstract simplicial complex can be restated as saying that every face of a face of a complex is itself a face of . The vertex set of is dened as V() = , the union of all faces of . The elements of thevertex set are called the vertices of the complex. So for every vertex v of , the set {v} is a face of the complex.The maximal faces of (i.e., faces that are not subsets of any other faces) are called facets of the complex. Thedimension of a face X in is dened as dim(X) = |X| 1: faces consisting of a single element are zero-dimensional,faces consisting of two elements are one-dimensional, etc. The dimension of the complex dim() is dened as thelargest dimension of any of its faces, or innity if there is no nite bound on the dimension of the faces.The complex is said to be nite if it has nitely many faces, or equivalently if its vertex set is nite. Also, issaid to be pure if it is nite-dimensional (but not necessarily nite) and every facet has the same dimension. In otherwords, is pure if dim() is nite and every face is contained in a facet of dimension dim().One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertexset of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspondto undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that donot have any incident edges.A subcomplex of is a simplicial complex L such that every face of L belongs to ; that is, L and L is asimplicial complex. A subcomplex that consists of all of the subsets of a single face of is often called a simplexof . (However, some authors use the term simplex for a face or, rather ambiguously, for both a face and thesubcomplex associated with a face, by analogy with the non-abstract (geometric) simplicial complex terminology. Toavoid ambiguity, we do not use in this article the term simplex for a face in the context of abstract complexes.)The d-skeleton of is the subcomplex of consisting of all of the faces of that have dimension at most d. Inparticular, the 1-skeleton is called the underlying graph of . The 0-skeleton of can be identied with its vertexset, although formally it is not quite the same thing (the vertex set is a single set of all of the vertices, while the0-skeleton is a family of single-element sets).The link of a face Y in , often denoted /Y or lk(Y), is the subcomplex of dened by

/Y := fX 2 j X \ Y = ?; X [ Y 2 g:Note that the link of the empty set is itself.

1

2 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

Given two abstract simplicial complexes, and , a simplicial map is a function f that maps the vertices of to thevertices of and that has the property that for any face X of , the image set f (X) is a face of .

1.2 Geometric realizationWe can associate to an abstract simplicial complex K a topological space |K |, called its geometric realization, whichis a simplicial complex. The construction goes as follows.First, dene |K | as a subset of [0, 1]S consisting of functions t : S [0, 1] satisfying the two conditions:

Xs2S

ts = 1

fs 2 S : ts > 0g 2 Now think of [0, 1]S as the direct limit of [0, 1]A where A ranges over nite subsets of S, and give [0, 1]S the inducedtopology. Now give |K | the subspace topology.Alternatively, let K denote the category whose objects are the faces of K and whose morphisms are inclusions. Nextchoose a total order on the vertex set of K and dene a functor F from K to the category of topological spaces asfollows. For any face X K of dimension n, let F(X) = n be the standard n-simplex. The order on the vertex setthen species a unique bijection between the elements of X and vertices of n, ordered in the usual way e0 < e1 < ...< en. If Y X is a face of dimension m < n, then this bijection species a unique m-dimensional face of n. DeneF(Y) F(X) to be the unique ane linear embedding of m as that distinguished face of n, such that the map onvertices is order preserving.We can then dene the geometric realization |K | as the colimit of the functor F. More specically |K | is the quotientspace of the disjoint union

aX2K

F (X)

by the equivalence relation which identies a point y F(Y) with its image under the map F(Y) F(X), for everyinclusion Y X.If K is nite, then we can describe |K | more simply. Choose an embedding of the vertex set of K as an anelyindependent subset of some Euclidean space RN of suciently high dimension N. Then any face X K can beidentied with the geometric simplex in RN spanned by the corresponding embedded vertices. Take |K | to be theunion of all such simplices.If K is the standard combinatorial n-simplex, then |K | can be naturally identied with n.

1.3 Examples As an example, let V be a nite subset of S of cardinality n + 1 and let K be the power set of V. Then K is called

a combinatorial n-simplex with vertex set V. If V = S = {0, 1, ..., n}, K is called the standard combinatorialn-simplex.

The clique complex of an undirected graph has a simplex for each clique (complete subgraph) of the givengraph. Clique complexes form the prototypical example of ag complexes, complexes with the property thatevery set of elements that pairwise belong to simplexes of the complex is itself a simplex.

In the theory of partially ordered sets (posets), the order complex of a poset is the set of all nite chains.Its homology groups and other topological invariants contain important information about the poset.

The VietorisRips complex is dened from any metric space M and distance by forming a simplex for everynite subset of M with diameter at most . It has applications in homology theory, hyperbolic groups, imageprocessing, and mobile ad hoc networking. It is another example of a ag complex.

1.4. ENUMERATION 3

1.4 EnumerationThe number of abstract simplicial complexes on n elements is one less than the nth Dedekind number. These numbersgrow very rapidly, and are known only for n 8; they are (starting with n = 0):

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 (sequence A014466 inOEIS).

1.5 See also KruskalKatona theorem

1.6 References[1] Lee, JM, Introduction to Topological Manifolds, Springer 2011, ISBN 1-4419-7939-5, p153

[2] Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991). Greedoids. Springer-Verlag. p. 9. ISBN 3-540-18190-3.

4 CHAPTER 1. ABSTRACT SIMPLICIAL COMPLEX

A geometrical representation of an abstract simplicial complex that is not a valid simplicial complex.

Chapter 2

Almost disjoint sets

In mathematics, two sets are almost disjoint [1][2] if their intersection is small in some sense; dierent denitions ofsmall will result in dierent denitions of almost disjoint.

2.1 DenitionThe most common choice is to take small to mean nite. In this case, two sets are almost disjoint if their intersectionis nite, i.e. if

jA \Bj

6 CHAPTER 2. ALMOST DISJOINT SETS

2.2 Other meaningsSometimes almost disjoint is used in some other sense, or in the sense of measure theory or topological category.Here are some alternative denitions of almost disjoint that are sometimes used (similar denitions apply to innitecollections):

Let be any cardinal number. Then two sets A and B are almost disjoint if the cardinality of their intersectionis less than , i.e. if

jA \Bj < :

The case of = 1 is simply the denition of disjoint sets; the case of

= @0

is simply the denition of almost disjoint given above, where the intersection of A and B is nite.

Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if theirintersection is a null-set, i.e. if

m(A \B) = 0:

Let X be a topological space. Then two subsets A and B of X are almost disjoint if their intersection is meagrein X.

2.3 References[1] Kunen, K. (1980), Set Theory; an introduction to independence proofs, North Holland, p. 47

[2] Jech, R. (2006) Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118

Chapter 3

Antimatroid

{a,b}

{a,b,c}

{a,c} {b,c}

{a} {c}

abcaba

acacbccacabcbcba

{a} {c}

{a,b} {b,c}

{a,b,c,d}

abcd

acbd

cabd

cbad

{a,b,c,d}

Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding pathposet.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by includingelements one at a time, and in which an element, once available for inclusion, remains available until it is included.Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible statesof such a process, or as a formal language modeling the dierent sequences in which elements may be included.Dilworth (1940) was the rst to study antimatroids, using yet another axiomatization based on lattice theory, andthey have been frequently rediscovered in other contexts;[1] see Korte et al. (1991) for a comprehensive survey ofantimatroid theory with many additional references.The axioms dening antimatroids as set systems are very similar to those of matroids, but whereas matroids are denedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are dened insteadby an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of greedoidsand of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids areequivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.

7

8 CHAPTER 3. ANTIMATROID

Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequencesin simulations, task planning in articial intelligence, and the states of knowledge of human learners.

3.1 DenitionsAn antimatroid can be dened as a nite family F of sets, called feasible sets, with the following two properties:

The union of any two feasible sets is also feasible. That is, F is closed under unions.

If S is a nonempty feasible set, then there exists some x in S such that S \ {x} (the set formed by removing xfrom S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent denition as a formal language, that is, as a set of strings dened from a nitealphabet of symbols. A language L dening an antimatroid must satisfy the following properties:

Every symbol of the alphabet occurs in at least one word of L.

Each word of L contains at most one copy of any symbol.

Every prex of a string in L is also in L.

If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s suchthat tx is another string in L.

If L is an antimatroid dened as a formal language, then the sets of symbols in strings of L form an accessible union-closed set system. In the other direction, if F is an accessible union-closed set system, and L is the language of stringss with the property that the set of symbols in each prex of s is feasible, then L denes an antimatroid. Thus, thesetwo denitions lead to mathematically equivalent classes of objects.[2]

3.2 Examples A chain antimatroid has as its formal language the prexes of a single word, and as its feasible sets the sets

of symbols in these prexes. For instance the chain antimatroid dened by the word abcd has as its formallanguage the strings {, a, ab, abc, abcd"} and as its feasible sets the sets , {a}, {a,b}, {a,b,c}, and{a,b,c,d}.[3]

A poset antimatroid has as its feasible sets the lower sets of a nite partially ordered set. By Birkhos rep-resentation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion)form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can beseen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroidfor a total order.[3]

A shelling sequence of a nite set U of points in the Euclidean plane or a higher-dimensional Euclidean spaceis an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplanein a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertexof the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid,called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with thecomplement of a convex set.[3]

A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v,the neighbors of v that occur later than v in the ordering form a clique. The prexes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

3.3. PATHS AND BASIC WORDS 9

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have beenremoved.

3.3 Paths and basic words

In the set theoretic axiomatization of an antimatroid there are certain special sets called paths that determine thewhole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If S is any feasible setof the antimatroid, an element x that can be removed from S to form another feasible set is called an endpoint of S,and a feasible set that has only one endpoint is called a path of the antimatroid. The family of paths can be partiallyordered by set inclusion, forming the path poset of the antimatroid.For every feasible set S in the antimatroid, and every element x of S, one may nd a path subset of S for which xis an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset.Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in thisunion is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to theantimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions oftheir proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroidif and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is thefamily of unions of subsets of P.In the formal language formalization of an antimatroid we may also identify a subset of words that determine the wholelanguage, the basic words. The longest strings in L are called basic words; each basic word forms a permutation ofthe whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partialorder. If B is the set of basic words, L can be dened from B as the set of prexes of words in B. It is often convenientto dene antimatroids from basic words in this way, but it is not straightforward to write an axiomatic denition of

10 CHAPTER 3. ANTIMATROID

antimatroids in terms of their basic words.

3.4 Convex geometriesSee also: Convex set, Convex geometry and Closure operator

If F is the set system dening an antimatroid, with U equal to the union of the sets in F, then the family of sets

G = fU n S j S 2 Fg

complementary to the sets in F is sometimes called a convex geometry, and the sets in G are called convex sets. Forinstance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean spaceinto which U is embedded.Complementarily to the properties of set systems that dene antimatroids, the set system dening a convex geometryshould be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in Sthat can be added to S to form another set in G.A convex geometry can also be dened in terms of a closure operator that maps any subset of U to its minimalclosed superset. To be a closure operator, should have the following properties:

() = : the closure of the empty set is empty. Any set S is a subset of (S). If S is a subset of T, then (S) must be a subset of (T). For any set S, (S) = ((S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. Theclosure operators that dene convex geometries also satisfy an additional anti-exchange axiom:

If neither y nor z belong to (S), but z belongs to (S {y}), then y does not belong to (S {z}).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchangeclosure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x y inthe partial order when x belongs to (S {y}). If x is a minimal element of this partial order, then S {x} is closed.That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universalset there is an element x that can be added to it to produce another closed set. This property is complementary tothe accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementaryto the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closedsets of any anti-exchange closure form an antimatroid.[4]

3.5 Join-distributive latticesAny two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound(the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid,partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted inlattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the correspondinglattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise fromantimatroids in this way generalize the nite distributive lattices, and can be characterized in several dierent ways.

The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice.For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x (Sxis the union of all feasible sets not containing x). Sx is meet-irreducible, meaning that it is not the meet of any

3.6. SUPERSOLVABLE ANTIMATROIDS 11

two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and sodoes not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often inmultiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal familyof meet-irreducible sets Sx whose meet is T ; this family consists of the sets Sx such that T {x} belongs to theantimatroid. That is, the lattice has unique meet-irreducible decompositions.

A second characterization concerns the intervals in the lattice, the sublattices dened by a pair of lattice elementsx y and consisting of all lattice elements z with x z y. An interval is atomistic if every element in it is thejoin of atoms (the minimal elements above the bottom element x), and it is Boolean if it is isomorphic to thelattice of all subsets of a nite set. For an antimatroid, every interval that is atomistic is also boolean.

Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimod-ular law that for any two elements x and y, if y covers x y then x y covers x. Translating this conditioninto the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element maybe added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x y and x z are both equal then they alsoequal x (y z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has booleanatomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducibledecompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositionsand boolean atomistic intervals.[5] Thus, we may refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a nite join-distributive lattice, and any nite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of nite join-distributive lattices is that they are graded(any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.[7] The antimatroid representing a nite join-distributive lattice can be recoveredfrom the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, andthe feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y suchthat y is not greater than or equal to x in the lattice.This representation of any nite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhos representation theorem under which any nite distributivelattice has a representation as a family of sets closed under unions and intersections.

3.6 Supersolvable antimatroidsMotivated by a problem of dening partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is dened by a totally ordered collectionof elements, and a family of sets of these elements. The family must include the empty set. Additionally, it musthave the property that if two sets A and B belong to the family, the set-theoretic dierence B \ A is nonempty, and xis the smallest element of B \ A, then A {x} also belongs to the family. As Armstrong observes, any family of setsof this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroidsthat this construction can form.

3.7 Join operation and convex dimensionIf A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, wecan form another antimatroid, the join of A and B, as follows:

A _B = fS [ T j S 2 A ^ T 2 Bg:

Note that this is a dierent operation than the join considered in the lattice-theoretic characterizations of antimatroids:it combines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to formanother set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.

12 CHAPTER 3. ANTIMATROID

Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of alanguage L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible setsthe unions of prexes of strings in L. In terms of this closure operation, the join is the closure of the union of thelanguages of A and B.Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure ofa set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (orequivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroidswhose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. Thepaths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimensionof an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworths theoremequals the width of the path poset.[8]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation canbe used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate perbasic word w, and make the coordinate value of a feasible set S be the length of the longest prex of w that is asubset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal tothe corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets isat most equal to the convex dimension of the antimatroid.[9] However, in general these two dimensions may be verydierent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

3.8 EnumerationThe number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. Forsets of one, two, three, etc. elements, the number of distinct antimatroids is

1, 3, 22, 485, 59386, 133059751, ... (sequence A119770 in OEIS).

3.9 ApplicationsBoth the precedence and release time constraints in the standard notation for theoretic scheduling problems maybe modeled by antimatroids. Boyd & Faigle (1990) use antimatroids to generalize a greedy algorithm of EugeneLawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal isto minimize the maximum penalty incurred by the late scheduling of a task.Glasserman & Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems.Parmar (2003) uses antimatroids to model progress towards a goal in articial intelligence planning problems.In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems thathe or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets ofconcepts that could be understood by a single person. The axioms dening an antimatroid may be phrased informallyas stating that learning one concept can never prevent the learner from learning another concept, and that any feasiblestate of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment systemis to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosenset of problems. In this context antimatroids have also been called learning spaces and well-graded knowledgespaces.[10]

3.10 Notes[1] Two early references are Edelman (1980) and Jamison (1980); Jamison was the rst to use the term antimatroid.

Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

3.11. REFERENCES 13

[3] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earliere.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for eciently listingall perfect elimination orderings of a given chordal graph.

[4] Korte et al., Theorem 1.1.[5] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.[6] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.[7] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.[8] Edelman & Saks (1988); Korte et al., Theorem 6.9.[9] Korte et al., Corollary 6.10.

[10] Doignon & Falmagne (1999).

3.11 References Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), Join-semidistributive lattices and convex ge-

ometries, Advances in Mathematics 173 (1): 149, doi:10.1016/S0001-8708(02)00011-7. Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047. Birkho, Garrett; Bennett, M. K. (1985), The convexity lattice of a poset,Order 2 (3): 223242, doi:10.1007/BF00333128. Bjrner, Anders; Ziegler, Gnter M. (1992), 8 Introduction to greedoids, in White, Neil, Matroid Appli-cations, Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp.284357, doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537

Boyd, E. Andrew; Faigle, Ulrich (1990), An algorithmic characterization of antimatroids, Discrete AppliedMathematics 28 (3): 197205, doi:10.1016/0166-218X(90)90002-T.

Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), Generating and characterizing the perfect elimina-tion orderings of a chordal graph (PDF), Theoretical Computer Science 307 (2): 303317, doi:10.1016/S0304-3975(03)00221-4.

Dilworth, Robert P. (1940), Lattices with unique irreducible decompositions, Annals of Mathematics 41 (4):771777, doi:10.2307/1968857, JSTOR 1968857.

Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

Edelman, Paul H. (1980), Meet-distributive lattices and the anti-exchange closure, Algebra Universalis 10(1): 290299, doi:10.1007/BF02482912.

Edelman, Paul H.; Saks, Michael E. (1988), Combinatorial representation and convex dimension of convexgeometries, Order 5 (1): 2332, doi:10.1007/BF00143895.

Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Prob-ability and Statistics, Wiley Interscience, ISBN 978-0-471-58041-6.

Gordon, Gary (1997), A invariant for greedoids and antimatroids, Electronic Journal of Combinatorics 4(1): Research Paper 13, MR 1445628.

Jamison, Robert (1980), Copoints in antimatroids, Proceedings of the Eleventh Southeastern Conference onCombinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, Con-gressus Numerantium 29, pp. 535544, MR 608454.

Korte, Bernhard; Lovsz, Lszl; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 1943,