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Sperner’s Lemma
Ralph Freese
Ralph Freese () Sperner’s Lemma Aug 17, 2011 1 / 23
Paul Erdos
Figure: Paul ErdosRalph Freese () Sperner’s Lemma Aug 17, 2011 2 / 23
Pál Erdos
Figure: Pál ErdosRalph Freese () Sperner’s Lemma Aug 17, 2011 3 / 23
Erdos Pál
Figure: Erdos PálRalph Freese () Sperner’s Lemma Aug 17, 2011 4 / 23
Paul Erdos
Figure: Paul ErdosRalph Freese () Sperner’s Lemma Aug 17, 2011 5 / 23
Sperner’s Lemma
S is a finite set with n elements.
P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 6 / 23
Sperner’s Lemma
S is a finite set with n elements.P(S) is the set of all subsets of S.
An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 6 / 23
Sperner’s Lemma
S is a finite set with n elements.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.
Problem: How big can an antichain in P(S) be?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 6 / 23
Sperner’s Lemma
S is a finite set with n elements.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 6 / 23
A Picture (Hasse Diagram)
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 7 / 23
A Picture (Hasse Diagram)
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{3}{2}{1}
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 7 / 23
A Picture (Hasse Diagram)
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{1,2,3}
{1,3} {2,3}
{3}{2}{1}
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ww
w w w
www
w w w
Ralph Freese () Sperner’s Lemma Aug 17, 2011 7 / 23
A Picture (Hasse Diagram)
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{1,2,3}
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{3}{2}{1}
{1,2} w
ww
w w w
www
w w w
Ralph Freese () Sperner’s Lemma Aug 17, 2011 7 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?
Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 8 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 8 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.
(nk
)is largest for k = dn/2e.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 8 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 8 / 23
P(S) with |S| = 4
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 9 / 23
P(S) with |S| = 4
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 9 / 23
P(S) with |S| = 4
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 9 / 23
P(S) with |S| = 4
v
v
v v v vv v v v v v
v v v v
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v v v vv v v vv v v v v v
v v v vv v v v v v
Ralph Freese () Sperner’s Lemma Aug 17, 2011 9 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Are there any antichains with more than( ndn/2e
)sets?
Theorem (Sperner)No!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 10 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Are there any antichains with more than( ndn/2e
)sets?
Theorem (Sperner)No!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 10 / 23
Sperner’s Lemma
S is a finite set with n elements. That is |S| = n.P(S) is the set of all subsets of S.An antichain in P(S) is a collections of subsets {S1, . . . ,St} of Ssuch that none is a subset of any of the others.Problem: How big can an antichain in P(S) be?Observations:
Sk = {A ∈ P(S) : |A| = k} is an antichain with(n
k
)= n!
k!(n−k)!
elements.(nk
)is largest for k = dn/2e.
Are there any antichains with more than( ndn/2e
)sets?
Theorem (Sperner)No!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 10 / 23
Partially Ordered Sets and Lattices
A set P together with a relation ≤ is called a (partially) orderedset if it satisfies:
x ≤ x (reflexive)if x ≤ y and y ≤ x , then x = y (antisymmetric)if x ≤ y and y ≤ z, then x ≤ z (transitive)
Example: P = P(S) with ≤ equal to ⊆.A lattice is an ordered set such that each pair x and y has a leastupper bound, x ∨ y , and greatest lower bound, x ∧ y .x ∨ y is also called the join of x and y .x ∧ y is also called the meet of x and y .Example: P = P(S) with ∨ equal to ∪ and ∧ equal to ∩.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 11 / 23
Partially Ordered Sets and Lattices
A set P together with a relation ≤ is called a (partially) orderedset if it satisfies:
x ≤ x (reflexive)if x ≤ y and y ≤ x , then x = y (antisymmetric)if x ≤ y and y ≤ z, then x ≤ z (transitive)
Example: P = P(S) with ≤ equal to ⊆.
A lattice is an ordered set such that each pair x and y has a leastupper bound, x ∨ y , and greatest lower bound, x ∧ y .x ∨ y is also called the join of x and y .x ∧ y is also called the meet of x and y .Example: P = P(S) with ∨ equal to ∪ and ∧ equal to ∩.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 11 / 23
Partially Ordered Sets and Lattices
A set P together with a relation ≤ is called a (partially) orderedset if it satisfies:
x ≤ x (reflexive)if x ≤ y and y ≤ x , then x = y (antisymmetric)if x ≤ y and y ≤ z, then x ≤ z (transitive)
Example: P = P(S) with ≤ equal to ⊆.A lattice is an ordered set such that each pair x and y has a leastupper bound, x ∨ y , and greatest lower bound, x ∧ y .
x ∨ y is also called the join of x and y .x ∧ y is also called the meet of x and y .Example: P = P(S) with ∨ equal to ∪ and ∧ equal to ∩.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 11 / 23
Partially Ordered Sets and Lattices
A set P together with a relation ≤ is called a (partially) orderedset if it satisfies:
x ≤ x (reflexive)if x ≤ y and y ≤ x , then x = y (antisymmetric)if x ≤ y and y ≤ z, then x ≤ z (transitive)
Example: P = P(S) with ≤ equal to ⊆.A lattice is an ordered set such that each pair x and y has a leastupper bound, x ∨ y , and greatest lower bound, x ∧ y .x ∨ y is also called the join of x and y .x ∧ y is also called the meet of x and y .
Example: P = P(S) with ∨ equal to ∪ and ∧ equal to ∩.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 11 / 23
Partially Ordered Sets and Lattices
A set P together with a relation ≤ is called a (partially) orderedset if it satisfies:
x ≤ x (reflexive)if x ≤ y and y ≤ x , then x = y (antisymmetric)if x ≤ y and y ≤ z, then x ≤ z (transitive)
Example: P = P(S) with ≤ equal to ⊆.A lattice is an ordered set such that each pair x and y has a leastupper bound, x ∨ y , and greatest lower bound, x ∧ y .x ∨ y is also called the join of x and y .x ∧ y is also called the meet of x and y .Example: P = P(S) with ∨ equal to ∪ and ∧ equal to ∩.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 11 / 23
Partially Ordered Sets and Lattices
Some ordered sets that aren’t lattices:
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 12 / 23
Partially Ordered Sets and Lattices
Some ordered sets that aren’t lattices:
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 12 / 23
Partially Ordered Sets and Lattices
Some ordered sets that aren’t lattices:
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 12 / 23
Automorphisms of Ordered Sets
An automorphism of an ordered set P is a one-to-one and ontomap σ : P → P such that x ≤ y in P iff σ(x) ≤ σ(y).
If G is a set of automorphisms closed under function compositionand inverses, then G is called a group.Problem: If P is a finite ordered set and G is a group ofautomorphisms of P, does P have a maximal-sized antichain Ainvariant under G; that is,
σ(A) = A for all σ ∈ G?
(σ(A) = {σ(a) : a ∈ A})
Theorem (Kleitman, Edelberg, Lubell)Yes!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 13 / 23
Automorphisms of Ordered Sets
An automorphism of an ordered set P is a one-to-one and ontomap σ : P → P such that x ≤ y in P iff σ(x) ≤ σ(y).If G is a set of automorphisms closed under function compositionand inverses, then G is called a group.
Problem: If P is a finite ordered set and G is a group ofautomorphisms of P, does P have a maximal-sized antichain Ainvariant under G; that is,
σ(A) = A for all σ ∈ G?
(σ(A) = {σ(a) : a ∈ A})
Theorem (Kleitman, Edelberg, Lubell)Yes!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 13 / 23
Automorphisms of Ordered Sets
An automorphism of an ordered set P is a one-to-one and ontomap σ : P → P such that x ≤ y in P iff σ(x) ≤ σ(y).If G is a set of automorphisms closed under function compositionand inverses, then G is called a group.Problem: If P is a finite ordered set and G is a group ofautomorphisms of P, does P have a maximal-sized antichain Ainvariant under G; that is,
σ(A) = A for all σ ∈ G?
(σ(A) = {σ(a) : a ∈ A})
Theorem (Kleitman, Edelberg, Lubell)Yes!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 13 / 23
Automorphisms of Ordered Sets
An automorphism of an ordered set P is a one-to-one and ontomap σ : P → P such that x ≤ y in P iff σ(x) ≤ σ(y).If G is a set of automorphisms closed under function compositionand inverses, then G is called a group.Problem: If P is a finite ordered set and G is a group ofautomorphisms of P, does P have a maximal-sized antichain Ainvariant under G; that is,
σ(A) = A for all σ ∈ G?
(σ(A) = {σ(a) : a ∈ A})
Theorem (Kleitman, Edelberg, Lubell)Yes!
Ralph Freese () Sperner’s Lemma Aug 17, 2011 13 / 23
Maximal-Sized Antichains
Let P a finite ordered set.
Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.
For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.
Theorem (Dilworth): L is a lattice. For C and D ∈ LC ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ B
C t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ B
C t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC
t t t t tt t t t t
D t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
Maximal-Sized Antichains
Let P a finite ordered set.Let L be the collection of all maximal-sized antichains in P.For A, B ∈ L, let A ≤ B if ∀a ∈ A ∃b ∈ B with a ≤ b.Theorem (Dilworth): L is a lattice. For C and D ∈ L
C ∨ D is the maximal elements of C ∪ D.C ∧ D is the miminal elements of C ∪ D.
t t t t tt t t t t
A ≤ BC
t t t t tt t t t t
D
t t t t tt t t t t
C ∧ D t t td t ttd td t td td
Ralph Freese () Sperner’s Lemma Aug 17, 2011 14 / 23
A Corollary to Dilworth’s Theorem
If A is a maximal-sized antichain of P and σ is an automorphism,the σ(A) is also a maximal-sized antichain.
σ “acts” on L.Since L is a finite lattice, it has a unique maximal (top) element, T .Clearly σ(T ) = T , so the theorem of Kleitman, et al., is true.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 15 / 23
A Corollary to Dilworth’s Theorem
If A is a maximal-sized antichain of P and σ is an automorphism,the σ(A) is also a maximal-sized antichain. σ “acts” on L.
Since L is a finite lattice, it has a unique maximal (top) element, T .Clearly σ(T ) = T , so the theorem of Kleitman, et al., is true.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 15 / 23
A Corollary to Dilworth’s Theorem
If A is a maximal-sized antichain of P and σ is an automorphism,the σ(A) is also a maximal-sized antichain. σ “acts” on L.Since L is a finite lattice, it has a unique maximal (top) element, T .
Clearly σ(T ) = T , so the theorem of Kleitman, et al., is true.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 15 / 23
A Corollary to Dilworth’s Theorem
If A is a maximal-sized antichain of P and σ is an automorphism,the σ(A) is also a maximal-sized antichain. σ “acts” on L.Since L is a finite lattice, it has a unique maximal (top) element, T .Clearly σ(T ) = T , so the theorem of Kleitman, et al., is true.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 15 / 23
Proof of Dilworth’s Theorem
For A and B ∈ L, let
AB = {a ∈ A : a ≥ b for some b ∈ B}AB = {a ∈ A : a ≤ b for some b ∈ B}
Since every element of P is comparable to something in A
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Ralph Freese () Sperner’s Lemma Aug 17, 2011 16 / 23
Proof of Dilworth’s Theorem
For A and B ∈ L, let
AB = {a ∈ A : a ≥ b for some b ∈ B}AB = {a ∈ A : a ≤ b for some b ∈ B}
Since every element of P is comparable to something in A
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Ralph Freese () Sperner’s Lemma Aug 17, 2011 16 / 23
A Corollary to Dilworth’s Theorem
Inclusion-exclusion: for sets C and D
|C ∪ D| = |C|+ |D| − |C ∩ D|
|C ∪ D|+ |C ∩ D| = |C|+ |D|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 17 / 23
A Corollary to Dilworth’s Theorem
Inclusion-exclusion: for sets C and D
|C ∪ D| = |C|+ |D| − |C ∩ D||C ∪ D|+ |C ∩ D| = |C|+ |D|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 17 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|
= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|
= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|
= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|
= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|
= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
The equations from before:
A ∨ B = AB ∪ BA and A ∧ B = AB ∪ BA
AB ∪ AB = A and BA ∪ BA = B
AB ∩ BA = AB ∩ BA = A ∩ B
AB ∩ AB = BA ∩ BA = A ∩ B
Using inclusion-exclusion:
|A ∨ B|+ |A ∧ B| = |AB ∪ BA|+ |AB ∪ BA|= |AB|+ |BA| − |AB ∩ BA|+ |AB|+ |BA| − |AB ∩ BA|= |AB|+ |AB|+ |BA|+ |BA| − 2|A ∩ B|= |AB ∪ AB|+ |AB ∩ AB|+ |BA ∪ BA|+ |BA ∩ BA| − 2|A ∩ B|= |A|+ |B|
Ralph Freese () Sperner’s Lemma Aug 17, 2011 18 / 23
A Corollary to Dilworth’s Theorem
Let k be the number of elements in a maximal-sized antichain.
Since both A ∨ B and A ∧ B are antichains both have at most kelements.But |A∨B|+ |A∧B| = |A|+ |B|. So |A∨B| = |A∧B| = k and soboth are in L.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 19 / 23
A Corollary to Dilworth’s Theorem
Let k be the number of elements in a maximal-sized antichain.Since both A ∨ B and A ∧ B are antichains both have at most kelements.
But |A∨B|+ |A∧B| = |A|+ |B|. So |A∨B| = |A∧B| = k and soboth are in L.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 19 / 23
A Corollary to Dilworth’s Theorem
Let k be the number of elements in a maximal-sized antichain.Since both A ∨ B and A ∧ B are antichains both have at most kelements.But |A∨B|+ |A∧B| = |A|+ |B|.
So |A∨B| = |A∧B| = k and soboth are in L.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 19 / 23
A Corollary to Dilworth’s Theorem
Let k be the number of elements in a maximal-sized antichain.Since both A ∨ B and A ∧ B are antichains both have at most kelements.But |A∨B|+ |A∧B| = |A|+ |B|. So |A∨B| = |A∧B| = k and soboth are in L.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 19 / 23
Back to Sperner
Let S be a set with n elements and let P = P(S).Let G be the group of all 1-1 maps from S to S. Of course σ ∈ Gacts on P by X 7→ σ(X ).
If X and Y ⊆ S both have k elements, there is a σ ∈ G withσ(X ) = Y .By Kleitman, Edelberg and Lubell’s Theorem, P(S) has amaximal-sized antichain A invariant under G. (If X ∈ A and σ ∈ Gthen σ(X ) ∈ A.)So if X ∈ A has k elements, then A contains all subsets with kelements. This (easily) implies A must consist of all subsets of Sof size k , for some k , which proves Sperner’s Lemma.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 20 / 23
Back to Sperner
Let S be a set with n elements and let P = P(S).Let G be the group of all 1-1 maps from S to S. Of course σ ∈ Gacts on P by X 7→ σ(X ).If X and Y ⊆ S both have k elements, there is a σ ∈ G withσ(X ) = Y .
By Kleitman, Edelberg and Lubell’s Theorem, P(S) has amaximal-sized antichain A invariant under G. (If X ∈ A and σ ∈ Gthen σ(X ) ∈ A.)So if X ∈ A has k elements, then A contains all subsets with kelements. This (easily) implies A must consist of all subsets of Sof size k , for some k , which proves Sperner’s Lemma.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 20 / 23
Back to Sperner
Let S be a set with n elements and let P = P(S).Let G be the group of all 1-1 maps from S to S. Of course σ ∈ Gacts on P by X 7→ σ(X ).If X and Y ⊆ S both have k elements, there is a σ ∈ G withσ(X ) = Y .By Kleitman, Edelberg and Lubell’s Theorem, P(S) has amaximal-sized antichain A invariant under G. (If X ∈ A and σ ∈ Gthen σ(X ) ∈ A.)
So if X ∈ A has k elements, then A contains all subsets with kelements. This (easily) implies A must consist of all subsets of Sof size k , for some k , which proves Sperner’s Lemma.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 20 / 23
Back to Sperner
Let S be a set with n elements and let P = P(S).Let G be the group of all 1-1 maps from S to S. Of course σ ∈ Gacts on P by X 7→ σ(X ).If X and Y ⊆ S both have k elements, there is a σ ∈ G withσ(X ) = Y .By Kleitman, Edelberg and Lubell’s Theorem, P(S) has amaximal-sized antichain A invariant under G. (If X ∈ A and σ ∈ Gthen σ(X ) ∈ A.)So if X ∈ A has k elements, then A contains all subsets with kelements.
This (easily) implies A must consist of all subsets of Sof size k , for some k , which proves Sperner’s Lemma.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 20 / 23
Back to Sperner
Let S be a set with n elements and let P = P(S).Let G be the group of all 1-1 maps from S to S. Of course σ ∈ Gacts on P by X 7→ σ(X ).If X and Y ⊆ S both have k elements, there is a σ ∈ G withσ(X ) = Y .By Kleitman, Edelberg and Lubell’s Theorem, P(S) has amaximal-sized antichain A invariant under G. (If X ∈ A and σ ∈ Gthen σ(X ) ∈ A.)So if X ∈ A has k elements, then A contains all subsets with kelements. This (easily) implies A must consist of all subsets of Sof size k , for some k , which proves Sperner’s Lemma.
Ralph Freese () Sperner’s Lemma Aug 17, 2011 20 / 23
Partition Lattices and Rota’s Problem
Figure: Gian-Carlo RotaRalph Freese () Sperner’s Lemma Aug 17, 2011 21 / 23
Partition Lattices and Rota’s Problem
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Ralph Freese () Sperner’s Lemma Aug 17, 2011 22 / 23
Partition Lattices and Rota’s Problem
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Gian-Carlo Rota: Does the largest sized antichain in the lattice ofpartitions of an n element set consist of all partitions with k block,for some k .
Ralph Freese () Sperner’s Lemma Aug 17, 2011 22 / 23
Partition Lattices and Rota’s Problem
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12|34 2|134 1|234 14|233|1244|12313|24
1234
1|2|3|4
34|1|2
Gian-Carlo Rota: Does the largest sized antichain in the lattice ofpartitions of an n element set consist of all partitions with k block,for some k .
Ralph Freese () Sperner’s Lemma Aug 17, 2011 22 / 23
Partition Lattices and Rota’s Problem
Theorem (Canfield)
Rota’s problem has a negative answer—at least if n > 1025.
ProblemWhat is Dilworth’s lattice for the lattice of equivalence relations of an nelement set?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 23 / 23
Partition Lattices and Rota’s Problem
Theorem (Canfield)
Rota’s problem has a negative answer—at least if n > 1025.
ProblemWhat is Dilworth’s lattice for the lattice of equivalence relations of an nelement set?
Ralph Freese () Sperner’s Lemma Aug 17, 2011 23 / 23