Ramsey Theory. Tibor Szabó Extremal Combinatorics, FU Berlin, WiSe pigeons. 1

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1 Ramsey Theory Tibor Szabó Exremal Combiaorics, FU Berli, WiSe Ramsey Theory Pigeohole Priciple Give pigeos i pigeoholes, here has o be a pigeohole wih a leas pigeos, ad a pigeohole wih a mos pigeos. 1 Of course he Pigeohole Priciple (PP) jus formulaes simple geeral properies of ay dece average -cocep: here should always be a isace ha is a leas he average ad a isace ha is a mos he average. For a formal proof, say of he firs saeme of he PP, oe ca oe ha he egaio is simply sayig ha all pigeoholes have sricly less ha, i.e. a mos 1 pigeos. This leads o a coradicio o all pigeos appearig i oe of hese ( ) pigeoholes, as 1 <. I he firs par of our course we will ake he Pigeohole Priciple o a whole ew level while sudyig boh he uaiaive ad he ualiaive aspecs of Ramsey heory. Ramsey s heorem for graphs.1 Two-colour Ramsey umbers for cliues Warm-up problem from sociology How may people ca be a a pary wihou hree muual frieds or hree muual sragers? Make a graph: verices = people, red edge = frieds, blue edge = sragers how large ca a wo-coloured complee graph wihou moochromaic riagles be? Aswer, par 1: a leas 5: red graph is C 5 Aswer, par : a mos 5: Suppose we have six verices, ad cosider he edges icide o he firs oe wlog (a leas) hree of hese are red (where 3 = 5 ; PP is used wih he 5 icide edges (pigeos) classified io classes (pigeoholes) accordig o heir color) if ay wo such edpois share a red edge red riagle, doe herefore he edpois of he hree red edges spa a blue riagle, doe 1 Or sayig he same more formally: if he elemes of a se Q are classified io pairwise disjoi subses (i.e. Q Q is he disjoi uio of he ses Q i, i = 1,... ), he here is a subse Q j wih Q j elemes ad here Q is a subse Q l wih Q e elemes. 1

2 Defiiio.1 (Ramsey umbers). Give s N, le R(s) be he miimum N such ha every red-blue colourig of he edges of K coais a subgraph isomorphic o K s he edges of which all have he same color (refered o as beig moochromaic (or m.c., for shor)). Observaios We have jus proved R(3) = 6 Upper boud proof: fidig a moochromaic cliue i a arbirary colourig Lower boud proof: cosrucio of a specific colourig wihou moochromaic cliues Ofe coveie o oly cosider red subgraph: cliues red cliues, idepede ses blue cliues Fiieess of R(4 for example is oally uclear a his poi Theorem. (Ramsey [], 1930). For every s N, R(s) is fiie. Philosophy: every large sysem, o maer how chaoic, coais ordered subsysems" Quiesseial Ramsey resul fid moochromaic subsrucures i large coloured srucures Ramsey: Briish logicia, primarily ieresed i exisece of R(s) Claim.3. For every s N, R(s) 4 s. Proof. Le = s, ad fix a arbirary red/blue edge-colourig c : E(K ) {red, blue} of K. We will fid a moochromaic K s. To his ed we firs will fid a seuece of verices v 1, v,..., v s V := V (K ), which is righ-moochromaic, by which we mea ha for ay fixed idex i = 1,,..., s 3, he edges goig from v i o a verex v j wih a larger idex j have he same color. I oher words for ay i = 1,,..., s 3, here exiss a color c (i) {red, blue}, such ha c(v i v j ) = c (i) for every j, i < j s. Oce we fid such a righ moochromaic seuece, we will be doe. Ideed, he PP provides us wih a subseuece v i1,..., v is 1 of legh s 3 = s 1, such ha c (i 1 ) = = c (i s 1 ) ad he he verices v i1,..., v is 1, ogeher wih he las verex v s form a moochromaic cliue of order s (i color c (i 1 )). So o complee he proof we jus eed o fid his log eough righ-moochromaic seuece. We do his i a uie greedy fashio, usig agai he PP. We will keep pickig he ex verex arbirarily from he se of verices sill uder cosideraio, he deleig all eighbours whose edges are coloured wih he less freuely appearig colour, ad oe ha we have a leas half of he verices remaiig. Formally, le us se S 0 := V ad for every i = 0, 1,..., s 3 do he followig. Give a se S i of size s i, we selec a arbirary verex i S i, ame i v i+1, ad le B i+1 ad R i+1 deoe he ses of hose eighbors of v i+1 i S i which are coeced o i via a blue ad a red edge, respecively. The obviously B i+1 + R i+1 = S i 1. We choose S i+1 o be he larger of B i+1 ad R i+1, so for is size we have Bi + R i s i 1 S i+1 = = s (i+1), as desired. To complee he proof we jus eed o check ha his process ca go o log eough, i.e. v s ca acually be seleced. For ha we eed S s 3 o be o-empy, which is he case sice S s 3 = s (s 3) = 8. (So i fac i he heorem we could have claimed he upper boud 4 s /8 isead.) Eve hough his upper boud is geig close o beig a ceury old, he order 4 s is sill esseially he bes kow. We will reur o he uesio of how good hese bouds are whe we discuss lower bouds i he ex secio; for ow we see a couple of geeralisaios.

3 Hugaria mahemaicias Paul Erdős ad George Szekeres came across he problem idepedely (see heir moivaio wo secios laer), ad obaied slighly beer uaiaive bouds. For he improveme oe ca observe ha he proof above was uie waseful i he sese ha we always followed greedily he immediaely bes opio, owards he larger moochromaic degree, ad he we compleely igored he fac ha oce we did ha i some color, i ha color i is eough o fid a cliue of oe smaller order. This makes he problem asymmeric afer he firs sep of he proof, because i he oher color we sill eed o fid a cliue of same order as before. To accommodae his asymmery, he followig defiiio is ecessary. Defiiio.4 ((o ecessarily symmeric) Ramsey umbers). Give s, N, le R(s, ) be he miimum N such ha every red-blue colourig of he edges of K coais eiher a red K s or a blue K. Observaios Swappig red/blue: R(s, ) = R(, s) R(s, 1) = 1, R(s, = s. The followig upper boud of Erdős ad Szekeres will be proved o he homework as a guided exercise. Theorem.5 (Erdős Szekeres [1], 1935). For every s, N, R(s, ) ( ) s+ s 1. I paricular, ( ) 4 s R(s) = O. s. Geeralizaio 1: Ramsey s heorem for ifiie graphs Wha happes if we colour he edges of a ifiie graph, isead of a large fiie graph? Ifiie graphs Verex se N, Edge se ( ) N Colour every edge red or blue Fiie moochromaic cliues I paricular, for ay N by cosiderig he resricio of he colourig o he edges bewee he firs R(, ) umbers, we are guaraeed o fid a moochromaic cliue of size. Thus we defiiely have arbirarily large moochromaic cliues Ifiie moochromaic cliues This is NOT he same as a ifiie moochromaic cliue These large fiie cliues ca be bouded ad far apar Quesio: Do we ge a ifiie moochromaic cliue? Theorem.6 (Ramsey [], 1930). For ay wo-colourig of ( N, here exiss a ifiie se S N for which ( S is moochromaic. Proof. Oe ca repea he verex selecio procedure i he proof of Claim.3 ifiiely ofe ad hece creae a ifiie righ-moochromaic seuece. The proof of his is ideical o he oe here wih he obvious adapaio ha S i = B i+1 R i+1 beig ifiie implies S i+1 beig ifiie. Ad he ifiie righ-moochromaic seuece gives rise o a ifiie moochromaic cliue (as a leas oe of he colors mus occur ifiiely may imes amog he c -values). Homework: ifiie Ramsey Theorem fiie Ramsey Theorem 3

4 .3 Geeralizaio : Mulicolour Ramsey umbers I may applicaios he relaio bewee people (or oher eiies) are o ecessarily biary. Afer all, here mus be more o huma (or oher) relaios ha love ad hae. For his reaso he followig defiiio arises uie aurally. Defiiio.7 (Mulicolour Ramsey umbers). Give iegers r ad 1,,..., r N, le R r ( 1,,..., r ) be he miimum N such ha for ay colourig of he edges of K wih colours from [r], here is some idex i for which here is a moochromaic K i of colour i. Formally, by a r-colorig of he edges we mea a fucio c : E(K ) [r]. Noe ha we had o forge our ice habi of usig acual colors i our colorig ad rerea o he (probably more borig ad defiiely less colorful) realm of amig our colors by iegers. This is purely for pracical purposes, as saemes abou more ha wo colors become uie cumbersome o wrie dow whe usig o oly red ad blue, bu also yellow, gree, orage, purple, ec... You ge he picure(!) Theorem.8. For ay r ad 1,,..., r N, R r ( 1,,..., r ) is fiie. Proof. Proof by iducio o r, he umber of colours. Base case, r =, is Theorem.. For he iducio sep, suppose r 3, ad we have umbers 1,,..., r. We will ake a large eough, he formula give laer i he proof, ad fix a arbirary r-colourig c of he edges of K. The idea is o go colorblid, combie he las wo colors ogeher ad use he fiieess of he Ramsey umbers for r 1 colors. Of course his will guaraee wha we wa oly i he firs r colours. I order o have wha we wa i he las wo colors as well, we will ask our (r 1)-color Ramsey umber o deliver a large eough cliue i he las colour, so we ca use ha o ake boh of he colorblided orgial colors. Le us ow formalize his idea. We defie colorig c : E(K ) [r 1] from c. Le c (xy) = r 1 if c(xy) = r ad c (xy) = c(xy) oherwise. By he iducio hypohesis, R r 1 ( 1,,..., r, R( r 1, r )) is fiie, ad we choose = R r 1 ( 1,,..., r, R( r 1, r )). Noe ha here we use ha we use ha we already ca assume he fiieess of he Ramsey umber for ay large value of cliue orders if he umber of colors is oly r 1. Now he defiiio of he Ramsey umber provides us a appropriae moochromaic cliue i oe of he r 1 colors. If his moochromaic cliue is i oe of he firs r colours, he we are doe, as we he have a moochromaic cliue of size i i colour i, 1 i r. Oherwise we have a cliue of size R( r 1, r ) ha uses he combied colour. We ow resore he origial colourig, so ha all of hese edges are coloured eiher r 1 or r. By defiiio of R( r 1, r ), we also fid he desired moochromaic cliue i his case. Remarks. Wha kid of upper boud does his give? Followig he argume i he proof, we ge R r ( 1,,..., r ) R( 1, R(, R( 3,... R( r 1, r )...))), Applyig Theorem.5 ad he simplificaio ha ( ) s+ s 1 < s+, his shows ha we have R r ( 1,,..., r ) r 1 +r I paricular, R r (,,..., )...+1 (ower of heigh r) Ca we do beer? 4

5 By spliig colours evely ad mergig hem simulaeously i he above argume, oe ca reduce he upper boud o a ower of heigh log r. I he homework you are asked o give a upper boud of he form r i i (which is much beer!). 3 Lower bouds for Ramsey s heorem Recall ha o lower boud R(s, ) oe eeds o provide a colourig of a large complee graph wihou a red moochromaic K s ad a blue moochromaic K. For example for R(3, 3) we were lucky o have he C 5 -cosrucio ha complemes our upper boud of 6 perfecly ad hece proves ha R(3, 3) = 6. The value of R(4, 4) is kow (i is 18) maily because we are agai lucky eough o have a icredibly ice colorig o 17 verices which does he deed. Sarig from s 5 however, i is uclear how o geeralize his cosrucio he righ way. Or raher, he obvious geeralizaio does o aymore mach he upper bouds we have available from our various PP-based argumes. For R(5, 5) all wha is kow is ha 43 R(5, 5) 48. The upper boud was improved from 49 o 48 jus recely (las Sprig), wih heavy use of compuer checkig. I is worhwhile o hik over wha such a proof mus deal wih. There are (48 ) > red/blue-colourigs of he complee graph o 48 verices. The program mus cosider all of hem ad verify ha hey all coai a moochromaic K 5. Now, here are abou paricles i he (observable) uiverse ad he age of he uiverse is hough of beig abou 10 6 aosecods. So every sigle paricle i he uiverse has o check a leas 10 3 of hese cases i every sigle aosecod of is exisece ad he hey have a chace o be fiished by ow... This idicaes he eormous umbers ivolved i his simple combiaorial problem ad maybe explais our fuiliy i solvig i. Ad i also idicaes ha he rece verificaio mus do somehig clever besides pure brue-force checkig. 3.1 A firs idea: Dese K s -free graphs The firs idea oe migh have for a cosrucio is o be greedy. This someimes works, greedy algorihms are ofe effecive i compuer sciece. Here oe could argue wih he followig heurisic. Heurisic. We eed wo K s -free graphs complemeig each oher, ha is ogeher hey should occupy all he ( edges of K. Le us firs focus o he red graph ad make sure ha i uses up as may of hese edges as possible, ad deal wih he blue graph laer. This approach leads us o a aural exremal graph heory problem, askig for he maximum umber of edges a K s -free graph o verices ca have. Le us see firs wha happes whe s = 3, ha is, i he case of riagle-free graphs. Afer some rial ad error wih examples of riagle-free graphs o a small umber of verices, oe covices oeself ha he complee biparie graph K, seems o be a riagle-free graph wih may edges. The resul ha ideed oe cao do beer, i.e. ha every graph wih e ( K, ) + 1 = + 1 edges does have a riagle, is oe of he firs heorems of Exremal Graph Theory. ( ) Theorem 3.1 (Mael, 1907). If G is K 3 -free he e(g) e K,. Proof. Cosider a verex w of maximum degree i a riagle-free graph G, i.e. le d(w) = (G) =:. Recall ha N(w) is he eighborhood of w, ad le us deoe by R(w) = V (G) \ N(w) he res. We boud from above he umber of edges of G by addig up all he degrees of verices i 5

6 R(w). Ideed, by addig up he degrees of verices i R(w) we accou for each edge of G a leas oce, sice G is riagle-free, hece N(w) coais o edge. Coseuely, e(g) d(v) ( ) ( ) = R(w) = ( ) = e K,, v R(w) v R(w) as reuired. Here we used ha R(w) = N(w) =, ad he maximized he uadraic fucio x ( x)x over he iegers. Remark. Whe addig up he degrees i R(w) we accoued for each edge bewee R(w) ad N(w) exacly oce, ad for each iside R(w) exacly wice. The reaso we did o worry so much because of his overcou is our firm belief i our cosrucio beig opimal. I he complee biparie graph here are o edges iside R(w), so if i is ideed opimal we do o lose ayhig by his esimaio. The cosrucio of complee biparie graphs easily geeralizes whe isead of K 3 we wa o forbid K s+1. The we ca ake a graph wih a verex se pariioed io s pars ilcude all edges bewee pars ad o edges iside he pars. These graphs are called complee s-parie graphs ad ca be paramerized by he sizes of is pars 1,..., s. Complee s-parie graphs do o coai K s+1, sice wo of he s + 1 verices of ay copy of a K s+1 would have o be i he same par (by he PP), bu verices i he same par are o adjace, coradicio. Amog complee s-parie graphs he mos edges are coaied i he oe where he pars are as eual as possible, so ay wo pars have sizes differig by a mos oe. Ideed, oherwise we ca move a verex from a bigger par o smaller par ad icrease he umber of edges. This complee s-parie graph o verices, where he differece bewee he size of ay wo pars is a mos 1, is called he Turá-graph ad is deoed by T,s. Turá has show i 1941 (ad we will show i a couple of weeks) ha he Turá graph T,s is ideed he K s+1 -free graph wih he mos umber of edges o verices. Le us ow reur o our origial problem of cosrucig a appropriae -colorig. As he red graph, we decided o ake he K s -free Turá graph T,s 1 which uses up he mos edges from K. Wha is he he blue graph? I is he disjoi uio of s 1 cliues of order roughly s 1. I order o esure ha he blue graph also has o K s, we beer make sure ha s 1 < s, ha is (s 1). I oher words, wih his mehod we ca cosruced Ramsey graphs o (s 1) verices, bu o more. Hece R(s, s) (s 1) + 1, prey paheic whe compared o he bes kow upper boud, which sads close o 4 s. 3. The righ idea: radom cosrucio The colorig of he previous subsecio is prey simple, ye i is surprisigly hard o improve. For a shor period of ime Turá himself believed his cosrucio o be opimal. Erdős massively desroyed his belief i 1947 via a eually simple, bu fudameally differe idea. Heurisic. We wa he same from he red ad he blue graph (hey should be K s -free). Their roles are symmeric. Each edge has as much reaso o be red ha o be blue. Le us choose he color of each edge uiformly a radom, idepedely from each oher. Theorem 3. (Erdős, 1947). R(, ) (1 o(1)) e. Proof. The idea of his proof is o prove he exisece of a large Ramsey colourig wihou acually preseig i. Colour each edge of K by red or blue wih probabiliy 1/, such ha hese radom choices are muually idepede of each oher. I oher words, our probabiliy space cosiss of he se of all red/blue-colourigs of E(K ) wih all colorigs beig eually likely. We wa o avoid a moochromaic K. So for each R ( ) [], i.e. each se R of verices, we defie E R be he eve ha he iduced subgraph of K o R is moochromaic. The probabiliy 6

7 ( ha E R happes is: P(E R ) = ( 1 )( ad we have ) such eves. The probabiliy ha here exiss a moochromaic K ca he be esimaed by he uio boud P K P (R K ) = ( ( ) 1 ) ( ( e ) ( 1 (. R ( [] ) R ( [] ) If his expressio is less ha 1, he here exiss a red/blue-colorig of E(K ) wihou a moochromaic K. Takig he -h roo ad rearragig we obai ha if < 1 e, he P( here is a m.c. K ) < 1. Therefore, here exiss a red/blue-colourig wihou a moochromaic K o = 1 e verices. I exiss o wih posiive probabiliy, or 99% probabiliy, bu wih absolue, 100% ceraiy, SURELY THERE IS ONE. Ad hece, R(, ) (1 o(1)) e, as claimed. Le us remark ha his proof i fac shows ha almos every colourig of a K o wo less verices is a good colourig. However, we cao explicily fid oe. (See he Cosrucive Combiaorics course ex semeser.) Recall where we sad: R(, ) 4. So boh bouds are expoeial ow, bu hey are sill very far apar. A relaively rece improveme (abou a decade old) by a facor (which is superpolyomial, if ever so slighly) is cosidered a grea breakhrough ad appeared i he Aals of Mahemaics. Bu here are o improvemes o he bases. I paricular, i would be a faasic advace o prove ha R(, ) < holds. I he above proof he use of probabiliy is o esseial, oe could simply cou bad colorigs amog all colorigs ad coclude ha here mus be a good oe lef eve afer akig ou all he bad oes. Ulimaely his is rue abou every saeme i discree probabiliy. However, he idea of iroducig radomess is a major paradigm shif. I direcs our aeio o he various ools of probabiliy heory, some of which would really be problemaic o say, o o meio fid, hrough jus couig. The improveme of he ex secio is a iiial sep i his direcio. 3.3 A wis o he mehod: improvig he cosa facor I is worhwhile o oe ha oe ca prove ha wih probabiliy edig o 1, he radom colorig will coai moochromaic cliues of order k, so i a way he crude aalysis hrough he uio boud is esseially bes possible. Usig some aleraios o he radom cosrucio however, we ca improve he Erdős lower boud by a cosa facor of. By he above, i his regime i is simply o aymore rue ha he radom colorig is a good oe, sill here is a good oe. Theorem 3.3. R(, ) = (1 o(1)) e. Proof. Like i he previous heorem, le us colour he edges of K uiformly a radom by eiher red or blue wih probabiliy 1. As we meioed before his proof, if we raise above wha we have worked wih i Theorem 3., i is ieviable ha wih overwhelmig probabiliy here will be (may) moochromaic K. Our pla is o desroy each of hese by deleig a verex from hem ad hope ha he remaiig wo-colored cliue, ow wihou ay moochromaic K, has reaied mos of he origial verices. I oher words, we eed o show ha he umber of moochromaic K is of smaller order ha he umber of verices. To his ed, le X be he radom variable ha euals he umber of moochromaic K s i his wo-colourig. To have a idea abou his seemigly complicaed radom variable, we express We do o do i here. Oe mus use he secod mome mehod 7

8 i as he sum of may simple oes ad apply a simple ye surprigly powerful geeral propery of expecaio of variables: is lieariy. For each -eleme se K, le X K deoe he idicaor radom variable of he eve ha K iduces a moochromaic K. The X = X K ad by he lieariy of expecaio E[X] = K ( [] ) E[X k ] = K ( [] ) P(X K ) = ( ( ) 1 ) (. K ( [] ) Therefore, here exiss a colourig c such ha he umber of moochromaic K s is a mos ( ) ( 1 (. Fix such a colourig ad delee oe verex from each moochromaic K. This gives us a red/blue-colorig o a leas ( ( ) 1 ) ( verices wihou ay moochromaic K. Hece R(k, k) > where we esimaed ( ( ) ) ( e ( ) 1 ( ( e ) (1) ). Subsiuig = e, we obai a red/blue-colorig o ( ) e = (1 o(1)) e verices. 3 This shows he promised lower boud o R(, ). Refereces [1] P. Erdős ad G. Szekeres, A combiaorial problem i geomery, Composiio Mahemaica (1935), [] F. P. Ramsey, O a Problem of Formal Logic, Proc. Lodo Mah. Soc. 30 (1930), To opimize he choice of (i erms of ) migh be difficul o do precisely, because of he biomial coefficie ivolved i he expressio. I is easy o see however (ad i is worhwhile o acually do!), ha subsiuig a cosa facor larger, say =, would make he expressio o he righ had side of (1) egaive. So e ε we are a he asympoical opimum. 8