Asymptotic Behaviorof Max clique in Gnn G Aik

let W Gri E size of the maximum clique in Gn

Tim W Gn x 21092N for larsen foranyfixed Whp

W Gri E 2T E lookn

w Gn Z 2 E logan

i e wlGnto

P z

Pfof upper bound First moment method

To show IP W Gn 7 CHE 1092N o

s

a c

E El

Howmany different sets Sof knocksTherate L in total

1P I SEEN Grits is a K Clive

IPL yea 9 Gris is a K cliques

C Egg IPS Gn Is Is a k din

Kkk

probability decreasesink

Entropyleringrows MK

akoLpl Wan 212th Han

p SEEN Ents is a k eliten

E ZKako K 2

Gk

I nk 2 KKIKO

FEE 45k

EE n 2 HELk

qzaezygEEF dktmsutxcilelgn.ge

I n 24k

E 2 I o

why this is called first monet method

1k of k digus in Gn y If GnEDisakdice

Ip SEID SisakdHe Ipl Tk 70

P Tk 21 Tkisintgenvalue

E to

lklztkkn.ITBasically to show IP Wlan 7 ko FEE toit suffices to show E k O

0pfof tonerbound okTo show Ipl WCGn Z 2 E 1092N 1Note that

IPL wlGn z K Elp Tk o

IPL Tk 21

Q Is EI Te o sufficit forconduct lpl Teal 1

A In general No

EI let X

fO TE

say to10100 q

FIX low E 4090 2

PM X213 2 15 Ipwbdm r.ee small probably mass at an enormouslybig

valuei e Typical value is notcloseto mean

Q what we do then

Need concentration

EI Var X ELM ELxD 2

401005 E Colo E2

10200 ECI E

EIxD2second moment method

For an integer valued Xn to show 1PM 01it suffices to show

Varl Xn o EEN

Lemme Paley Zygmund inequalityLet X Zo be a random variable Then forany0 41

IP X c Eun t92E H2E a

Remarks let o_0 Then

IP X o zETD

Lectu I stop heTEETHE 1 if VIf oc

Lecture1 stopshere1 4 1

Pfoflemma

EEN EL X IS XECENTIFTEIL Is x CEHE CELX t EE XLIX CECH

I c ETX E EEXLIX CEE xD

TEXT dE cE

JEW xzcExTC c 2 xD E ELK IPC x CECH

Backto our original goal of shag

pl Tk 70 I for K K E Logan

It suffices toshow Van TeIEEE OCD

VarlTk Van 19Gn is a K clique 3

sq.FIGV 19Gnts7isakckgu3.1SGn s9isakdio

CoV 19antsisak ckgu3 ISGnsksakdic.at

slsHs

Cal follows because 9 is IES A Icij DES

9 if ISAs't EICorkill Cov _o

EX'D I LFLISGnts3isakdige346n sbisak.dkEITHER

1951122

ENDISHAK

whenxif20 I IPS both Grits and ants are k cliquesIs 22

HHS'tK

let f Isas't

s

IPSboth Grits and Grisi are K dyes

E 111 LEHI KHL

214 IE

How many such Sands with ISHS'fk If

There are E I III

e Kelli LETt

entropy term prob term

decreases in f grows in ldominated by e't which maximizes the summand

A Hypergeometric trick

EEE K El

Vantic ke IIe 211Et IE t C

12 PCH _ewhere Hn Hypergeo n k k

VagEL2

and bees if we draw191815122 nboe.es k bees outofthewww0wheeS1S1ayLFLf4snsYDkheds replacent

two indeplettcopesz 1pct e 2k

EL zkHk19h22E EE zkHk 113911 03

11394 03 µgn klCntn n D n KH

I E I 1 Ct FtN

Z 1 KE IEnKH

I k nti FEI 21as K E91092N

It remains to show E 2k I

Lemmaan Hoeffdy's lemma

Bin K dominates Hypergeolnskik in

the order of convex functions i e if BNBinds'T

EL ftHD E EE fCBD forolecux fHwy Bernanke HE Binlk EkUsing this lemma

id

EL 2 E EE 214 BinderK

EL z.EE3iJ

fELzEBgk

Intl Ei

ft HEGE IlE e

2124

as K2 2 1 En

1

Tosummarize we hateshown that

Van Tx

EEE o

Paley 9mMdpp Tk o I i e

IP W Gn 2,4 212 Ekka

k 5 In fact we show there are many Cligasofsize 2 E 1092N

EL k 2 ElKzK

2k

nNlk N p

Using Chebyshev's inequality

Pf Tn Z Ct d ETDT Y

Vastly VarinVa LENNIE 0

LEED Tz yteth VIETH

oc l

Tn z 4 och Etten whpi e there are super polynomially many chges

office k E loganUnfortunately the best polyve the altaronly guarantees to find a clique ofsize l E 10929

whp

plans Finish Grimmette McDiarmid's algorithmin next lecture