Highly connected coloured subgraphs via the Regularity Lemma

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1 Highly coeced coloued subgaphs via he Regulaiy Lemma Hey Liu 1 Depame of Mahemaics, Uivesiy College Lodo, Gowe See, Lodo WC1E 6BT, Uied Kigdom Yuy Peso 2 Isiu fü Ifomaik, Humbold-Uivesiä zu Beli, Ue de Lide 6, D Beli, Gemay Absac Fo ieges,, s, k N, k ad s, le m,, s, k be he lages i ode k-coeced compoe wih a mos s colous oe ca fid i ay -colouig of he edges of he complee gaph K o veices. Bollobás asked fo he deemiaio of m,, s, k. Hee, bouds ae obaied i he cases s = 1, 2 ad k = o, which exed esuls of Liu, Mois ad Pice. Ou echiques use Szemeédi s Regulaiy Lemma fo may colous. We shall also sudy a simila quesio fo bipaie gaphs. Key wods: Regulaiy Lemma, Gaph Ramsey umbes, k-coeced gaphs 1. Ioducio ad esuls Fo basic defiiios fom gaph heoy we efe he eade o [3]. Le,, s, k N wih s ad k. Give a gaph G = V, E wih V G =, ad a -colouig of is edges, f : EG [], defie Mf, G,, s, k := max{ V H : H G, feh s ad H is k-coeced}. Tha is, Mf, G,, s, k is he ode of he lages k-coeced subgaph H i G whose edges ae coloued wih a mos s diffee colous i he -colouig f. Le mg,, s, k := mi{mf, G,, s, k}. f I he case G = K, we wie Mf,,, s, k ad m,, s, k especively. The quesio of deemiig m,, s, k i is full geealiy was fis posed by Bollobás [2]. I paicula, Bollobás ad Gyáfás [4] cojecued ha m, 2, 1, k = 2k + 2 fo > 4k 1. They poved his fo k = 2, ad hey also showed ha m, 2, 1, k 62k 3 fo 16k 22 ad k 2. Fuhe paial esuls of he cojecue wee subsequely poved by Liu, Mois ad Pice. They poved he cojecue fo k = 3 [7], ad ha m, 2, 1, k = 2k + 2 fo 13k 15 [8]. Also i [8], hey sudied m,, 1, k fo geeal. They cojecued ha, give ad k wih 3, hee exiss 0 = 0, k such ha m,, 1, k is appoximaely equal o k+1 1, if 1 is a pime powe, fo evey 0. They poved he case = 3 wih 0 = 480k. The quesio becomes much moe hade o sudy whe oe looks fo mulicoloued k-coeced compoes, i.e. s 2. I a subseque pape [9], Liu, Mois ad Pice coduced fuhe eseach, deemiig moe values ad bouds fo m,, s, k wih s 2. I he wo papes [8, 9], Liu e al. poved amog ohe facs he followig lowe bouds fo,, k N ad 3: m,, 1, k 1 11k2 k, addesses: hey.liu@ucl.ac.uk Hey Liu, peso@ifomaik.hu-beli.de Yuy Peso 1 Reseach suppoed i pa by he Euopea Newok Pheomea i High Dimesios, FP6 Maie Cuie Acios, RTN, Coac MCRTN Reseach suppoed by GIF ga o. I /2005. Pepi submied o Elsevie Jue 5, 2009

2 m,, 2, k k + 2k + 1. Moeove, i he fis case, whe 1 is a pime powe, hey showed a uppe boud o m,, 1, k is k+1 1 +, ad i he secod case, whe + 1 is a powe of 2, a uppe boud o m,, 2, k is I he poofs of he above lowe bouds, Liu e al. used Made s Theoem [10]. Bu, hese lowe bouds ae good oly fo k = o. We will sudy asympoic lowe bouds o m,, s, k whe is lage. Usig ideas fom [8, 9] ad he may-colous vesio of Szemeédi s Regulaiy Lemma [12], we exed he above esuls o he subliea case: whe k = o ad hece, moe cosise wih he wo afoemeioed cojecues, showig ha asympoically, coecedess should play a lesse ole. Namely, ha mulicoloued compoes will coai highly coeced subgaphs of coeced coloued gaphs of liea ode. Moe pecisely, we shall pove he followig mai esuls. Theoem 1. Fo evey γ 0, 1 4,, N wih 3, hee exis ieges N 0 T 0 = T 0 γ, such ha, fo all N 0, m,, 1, 1 4γ T 0 1 6γ 1. = N 0 γ, ad I paicula, fo fixed, ad k = o, m,, 1, k powe. 1 o, wih equaliy if 1 is a pime Theoem 2. Fo evey γ 0, 1 4,, N wih 3, hee exis ieges N 0 T 0 = T 0 γ, such ha, fo all N 0, m,, 2, 1 4γ T γ + 1. = N 0 γ, ad I paicula, fo fixed, ad k = o, m,, 2, k 4 +1 o, wih equaliy if + 1 is a powe of 2. Bollobás ad Gyáfás [4] oed ha evey 2-coloued complee gaph o veices coais a 16 - coeced subgaph o a leas 4 veices. Thus, he above heoems may also be see as a sep i his diecio. Wih he same echiques, we ae also able o paially pove aohe cojecue of Liu e al.: Cojecue 2 of [8]. Fo his, i is moe coveie if we defie he aalogous fucio o m,, s, k fo bipaie gaphs. Fo,,, s, k N wih s ad k, ad a -colouig f : EK, [], defie M bip f,,,, s, k := max{ V H : H K,, feh s ad H is k-coeced}, m bip,,, s, k := mi{m bip f,,,, s, k}. f The cojecue he saes ha, povided, k, we have m bip,,, 1, k + idepede of k. We shall pove he followig paial esul. ad so Theoem 3. Fo evey γ 0, 1 34,,, N, wih 2 ad, hee exis ieges N 0 = N 0 γ, ad T 0 = T 0 γ, such ha, fo all N 0, + 3γ + m bip,,, 1, 1 4γ T I paicula, fo fixed, ad k = o, we have m bip,,, 1, k = + o. This pape will be ogaised as follows. I Secio 2, we shall discuss esuls elaed o he egulaiy lemma which we will equie o pove Theoems 1 ad 2. We he explai, i Secio 3, how we ca use hese esuls o pove he wo heoems. We epea his pocedue fo he bipaie gaphs sceaio. I Secio 4, we shall discuss bipaie egulaiy lemmas, ad he deduce Theoem 3. Fially, we shall discuss ope poblems i Secio 5. 2

3 2. Tools: Regulaiy Lemmas I his secio, we shall discuss he cocep of ε-egulaiy fo gaphs ad he celebaed Szemeédi s Regulaiy Lemma [12]. Fo fuhe deails, see he excelle suvey of Komlós ad Simoovis [6]. Fo a gaph G = V, E ad wo disjoi subses A, B of V, we wie EA, B o deoe he se of he edges fom E ha iesec boh A ad B. We se ea, B := EA, B. We wie G[A, B] fo he bipaie subgaph of G iduced by A ad B, ie: G[A, B] has veex classes A ad B, ad edge se EA, B. We ofe call A, B a pai wihou wiig G explicily whe i is clea fom he coex o if i is o impoa. Le ε 0, 1. We defie a pai V 1, V 2 a bipaie gaph wih veex classes V 1 ad V 2 o be ε-egula, if fo evey U i V i wih U i ε V i, i = 1, 2, he followig iequaliy holds: dv 1, V 2 du 1, U 2 < ε, whee dx, Y := ex,y X Y is he edge desiy of he pai X, Y. Noe ha if V 1, V 2 is ε-egula, he i is also ε -egula fo ay ε ε, 1. The pai V 1, V 2 is said o be ε, d-egula, if i is ε-egula wih dv 1, V 2 d. The followig well-kow lemmas make ε-egula pais vey useful i applicaios. Lemma 4 Facs 1.3, 1.4 of [6]. Give 0 < 2ε < η < 1, le V 1, V 2 be a ε-egula pai wih desiy η. The, {x V1 : Γx η ε V 2 } ε V 1, {x V 1 : Γx η + ε V 2 } ε V 1, {x, y : x, y V1, Γx Γy η ε 2 V 2 } 2ε V 1 2, {x, y : x, y V1, Γx Γy η + ε 2 V 2 } 2ε V 1 2. Noe ha x, y is a odeed pai. Lemma 5 Slicig Lemma; Fac 1.5 of [6]. Le ε, α 0, 1 wih ε < α, ad V 1, V 2 be a ε-egula pai. If U i V i ad U i α V i fo i = 1, 2, he U 1, U 2 is ε -egula, whee ε = max ε α, 2ε. Fo a give gaph G, he paiio V G = V 0 V 1 V 2 V of is veex se is said o be ε-egula if he followig codiios hold: V 0 ε V G, V i = V j, fo evey 1 i, j, ad all bu a mos ε 2 pais V i, V j ae ε-egula, 1 i < j. The classes of he paiio ae called cluses, ad V 0 is he excepioal se. Szemeédi s Regulaiy Lemma [12] he says ha, give ε 0, 1 ad a iege 0, we ca fid ieges N 0 = N 0 ε, 0 ad T 0 = T 0 ε, 0 such ha, fo evey gaph G o a leas N 0 veices, V G admis a paiio io +1 classes, fo some 0 T 0, which is ε-egula. Roughly speakig, his says ha ay gaph of sufficiely lage ode ca be appoximaed by a mulipaie gaph wih a bouded umbe of equal classes, whee he disibuio of he edges bewee mos pais of classes is, i some sese, as i a adom gaph. Hee, we shall uilise a saighfowad geealisaio of he oigial poof of Szemeédi: he maycolous egulaiy lemma. Theoem 6 May-colous egulaiy lemma; Theoem 1.18 of [6]. Fo evey ε 0, 1 ad, 0 N, hee exis ieges N 0 = N 0 ε,, 0 ad T 0 = T 0 ε,, 0 such ha he followig holds. Evey gaph G = V, E wih V N 0, whose edges ae -coloued: E = E 1 E, admis a paiio of is veex se: V = V 0 V 1 V, fo some 0 T 0, which is ε-egula simulaeously wih espec o evey subgaph G i = V, E i, i []. Fially, we will eed he oio of a cluse gaph of a gaph G. Le G be a gaph o veices, ad V G = V 1 V be a paiio of is veex se. Fo 0 < ε < η < 1 we hik of η as beig much lage ha ε, bu much smalle ha 1, defie a ew gaph Rη, he cluse gaph o educed 3

4 gaph, whee V Rη = [], ad ij ERη if V i, V j is ε, η-egula. Oe also someimes wies {V 1,..., V } fo V Rη. The emaiig udelyig gaph G is he he subgaph of G, whee V G = V G, ad xy EG if xy EG ad x V i, y V j, whee ij ERη. Tha is, we keep a edge fo G if he pai ha i belogs o is ε, η-egula. If i he oigial egulaiy lemma, we le 0 = 1 ε ad V G be sufficiely lage, he G admis a ε-egula paiio. If η is as above, we ca obai he gaph Rη fom G igoig he excepioal se o 1 ε veices, ad he gaph G. The, i G, we have disegaded a mos ε + ε 2 + ε η 2 / + < η2 edges fom G. Thus, fo small η, he gaph G is almos ou oigial gaph G. We ca similaly apply his fo he may-colous vesio of he egulaiy lemma, ad ge ha fo evey colou i [], he coespodig gaph G i is close o he gaph G i. I us ou ha, if we ca fid a compoe i Rη of ode c 2, he udelyig subgaph G of G coais a subgaph H o oughly a leas c veices, which oughly becomes η -coeced afe deleig a mos εc veices fom i. Lemma 7 Tee decomposiio lemma. Le ε 0, 1, ad suppose ha he gaph G has a paiio: V G = V 1 V, whee V i = m fo evey i. Fo 3ε < η < 1, le Rη be he cluse gaph of his paiio. If Rη coais a compoe of ode c 2, he G coais a η 3εm-coeced compoe o a leas 1 εcm veices. Poof. Le C be a compoe i Rη of ode c 2. Fix ay spaig ee T of C, ad assume wihou loss of geealiy ha V T = V C = {1,..., c}. The udelyig subgaph H of G ha coespods o T has V H = V 1 V c, ad xy EH if xy EG ad x V i, y V j, whee ij ET. We shall show ha, by deleig a mos εm veices fom each V i, i [c], we will ge a subgaph H of H which is η 3εm-coeced. We poceed as follows. Le L 1 be he leaves of T. Fo j > 1, le L j be he leaves of T j 1 i=1 L i defied iducively. We have a paiio V T = L 1 L p. Noe ha L p = 1 o 2, ad if p = 1, he L p = 2. Now, u he followig algoihm. Sep 1. If p = 1, he poceed o Sep 2. Ohewise, p > 1. I his case, ake a veex of L 1, say V i. I has exacly oe eighbou i T L 1, say V j. Sice V i, V j is ε, η-egula, by Lemma 4, we may delee all veices fom V i wih a mos η εm eighbous i V j, ad obai V i V i wih V i 1 εm. Now, disegad he veex V i fom T, ad epea his pocedue o evey veex of L 1. The, epea he whole pocedue successively o L 2, L 3,..., L p 1. Sep 2. We ae lef wih L p. If L p = 2, le L p = {V k, V l }. By Lemma 4, we may delee all veices fom V k wih a mos η εm eighbous i V l, ad all veices fom V l wih a mos η εm eighbous i V k. We obai V k V k ad V l V l wih V k, V l 1 εm. If L p = 1, le L p = {V k }. Take a abiay, fixed eighbou of V k i T, say V l, ad similaly delee he veices fom V k wih a mos η εm eighbous i V l. We obai V k V k wih V k 1 εm. Fo evey i [c], we have ow deleed a mos εm veices fom V i, obaiig V i V i. Le H be he emaiig subgaph of H. The, V H 1 εcm. We claim ha H is he equied η 3εmcoeced subgaph. We shall pove a soge asseio: deleig ay η 3εm veices fom evey V i does o discoec H. So, delee such a se of veices, le V i V i be he emaiig subses, i [c], ad le H be he emaiig subgaph of H. We wa o show ha H is coeced. We fis show ha he pai V k, V l is coeced. I suffices o show ha, fo evey x V k ad y V l, x is coeced o y. Obseve ha he miimum degee of he pai V k, V l is a leas η 2εm. So, if X = Γ H x V l ad Y = Γ Hy V k, he X, Y εm. I ow suffices o show ha EX, Y. Sice V k, V l is ε, η-egula, we have dx, Y > dv k, V l ε η ε > 0, which implies EX, Y. Now, ay V q V T \ {V k } is coeced o V k by a uique pah i T, say, V q1 V q, whee q 1 = q ad q = k. I is easy o see fom he algoihm ha, fo evey 1 s <, evey veex of V q s has a leas η 2εm η 3εm = εm eighbous i V q s+1. Hece, fo ay V, V V T which may be he same ad ay u V, v V, u ad v ae coeced o some u, v V k by H, especively. The lemma follows, sice u ad v ae coeced by V k, V l, so ha u ad v ae coeced by H. 4

5 3. Poofs of Theoems 1 ad 2 To obai Theoems 1 ad 2, we will geealise he followig esuls. Theoem 8 Theoem 11 of [8]. Le, N, wih 2. The m,, 1, 1 1. Theoem 9 Theoem 16 of [9]. Le, N, wih 3. The m,, 2, Ou mai goal will be o elax he em m,, s, 1 o mg,, s, 1, fo s = 1, 2, ad 3 i boh cases. Hee, G will be a almos complee gaph - i should miss a mos γ 2 edges, whee is he ode of G ad γ > 0 is small. A lemma of [8] ha we will eed is he followig. Lemma 10 Lemma 9 of [8]. Le m, N ad c [0, 1]. If G is a bipaie gaph wih pa-sizes m ad, ad eg cm, he G has a compoe of ode a leas cm +. We ae ow eady o pove he ew vesios of Theoems 8 ad 9. Theoem 11. Le γ 0, 1 ad, N, 3. Le G be a gaph o veices wih eg 2 γ 2. The mg,, 1, 1 1 9γ 2 1. Poof. Fix a -colouig of EG. We fis cosuc a bipaiio V G = V 1 V 2 such ha V 1, V 2 3 ad EV 1, V 2 has o edges of colou 1. Coside all he compoes of G i colou 1 icludig isolaed veices, if ay. Le hese have veex ses C 1,..., C p wih C 1 C 2 C p, ad p 2 we ae 1 2 clealy doe if p = 1. If C 1 1, he he heoem holds, so assume ha C 1 <. Now, ake V 1 = i=1 C i, V 2 = p i=+1 C i, whee {1,..., p 2} is he uique iege such ha 1 i=1 C i < 3 ad i=1 C i 3. Oe ca he easily check ha V 1, V 2 3. Now, ev 1, V 2 V 1 V 2 γ 2, ad hee exis a leas ev1,v2 edges i EV 1, V 2 of he same colou. Bu he Lemma 10 asses he exisece of a moochomaic compoe of ode a leas V 1 V 2 γ V 1 V 2 1 9γ We emak ha Theoem 11 beaks dow fo = 2, as he followig example of a 2-colouig o a gaph G o veices, ad missig γ 2 edges, shows. Take hee disjoi veex ses V 1, V 2, V 3 fo V G, whee V 1 = V 2 = γ ad V 3 = 1 2 γ. Colou he edges wihi V 1 ad bewee V 1 ad V 3 blue, hose wihi V 2 ad bewee V 2 ad V 3 ed, hose wihi V 3 abiaily, while bewee V 1 ad V 2 ae o-edges of G. The, i is easy o see ha his gives mg, 2, 1, 1 1 γ, showig ha Theoem 11 does o hold eve if we eplace he cosa 9 2 by ay ohe cosa. A geealisaio of Theoem 9 we will eed is as follows. Is poof is simila o he poof of Theoem 9. I is, howeve, moe echically ivolved, ad uses some addiioal ideas. Theoem 12. Fo evey γ 0, 1 ad, N, 3, hee exiss a δ = δγ, wih 0 < δ < γ such ha, if G is a gaph o veices wih eg 2 δ 2, he mg,, 2, γ +1. Poof. Le γ 0, 1 ad iege 3 be give. We will show how o choose 0 < δ = δγ, < γ o saisfy he heoem as we poceed hough he poof. Fix a -colouig of EG ad le H be a lages coeced moochomaic subgaph of G. Assume ha H is of colou 1. We se A = V H ad A = c1 γ +1. If c 4 he he heoem holds. So assume ow ha c < 4. Le B = V G \ A. The, o edge i G[A, B] has colou 1. Thus, hee exiss a colou, say 2, which occus a leas A B δ2 1 imes i G[A, B]. Le B 2 B be he veices of B which sed a edge of colou 2 o A. If B 2 4 c1 γ +1, he he se A B 2 is coeced by colous 1 ad 2, ad A B 2 41 γ +1, 5

6 so we ae doe. Thus, we may assume ha B 2 < 4 c1 γ +1, ad we ca choose a se B 2 such ha B 2 B 2 B ad B 2 = 4 c1 γ +1. Now coside he bipaie subgaph H 2 of G[A, B 2] whose edges ae of colou 2. We would like o choose δ so ha eh 2 A saighfowad bu edious calculaio yields A B δ2 + 1 c1 γ eh 2 = 1 1 γ Hece, we would like o hold. This is equivale o + 1 c1 γ 1 γ + 1 c 4 c 1 A B 2. 1 δ + 12 δ c1 γ δ c1 γ c 1 4 c 1 A B 2. cγ1 γ Povided ha δ 1 c1 γ 9, Theoem 11 gives δ Hece, if δ γ 2 2 as well, he 2 holds, ad so, 1 also holds. Now, applyig Lemma 10 o H 2, we have a coeced moochomaic subgaph o a leas + 1 c 4 c 1 A + B 2 = + 1 c 41 γ 4 c veices. Because H was chose o be a lages moochomaic subgaph i G, we obai + 1 c 41 γ 4 c c1 γ, + 1 ad if oe solves his iequaliy fo c see he poof of Theoem 9 i [9], agai a saighfowad calculaio yields c 1. We will oly eed c 2 fo lae. Nex, we aim o show ha a lage umbe of veices of B sed edges of a leas wo diffee colous o A. Suppose ha we have β veices of B which sed edges of exacly oe colou o A. The, a leas β 1 veices of B sed edges of oe paicula colou o A. Le B 3 be such a se of veices, wih B 3 β 1. The, agai by Lemma 10, we have a moochomaic compoe of ode a leas A B 3 δ 2 β + 1 A + B A B 3 1c1 γ δ2 1 δ + 12 βc1 γ c 2 1 γ 2 A. We will have a coadicio agais he maximaliy of H if β + 1 1c1 γ δ2 1 δ + 12 βc1 γ c 2 1 γ 2 > 0, ad solvig his as a quadaic i β, we ge β > δ2 1 2c1 γ + 1 δ c 2 1 γ + 4δ. So, we have 2 β δ2 1 2c1 γ + 1 δ c 2 + 4δ. 3 1 γ 2 Also, hee ae a mos δ 2 A = δ + 1 c1 γ 6 4

7 veices i B which have o eighbous i A. So, fom 3 ad 4, he umbe of veices i B sedig edges of a leas wo diffee colous o A is a leas B δ2 A β c1 γ δ c1 γ δ2 1 2c1 γ 1 δ c 2 1 γ 2 + 4δ. 5 We would like he quaiy o he igh of 5 o be a leas +1 c1 γ +1. This is ue if ad oly if afe some calculaio δ + 12 γ 2c1 γ + 1 δ c 2 1 γ 2 + 4δ. This holds if boh δ c1 γ γ 3 ad ae saisfied. So sufficiely ecallig ha c 2, we ca le δ 1 δ c 2 1 γ 2 + 4δ γ 3 4γ1 γ 16γ 2 1 γ ad δ γ 2. 6 I follows ha, akig δ which saisfies 6 as well as δ 1 9 ad δ γ 2 2 ad usig he pigeohole piciple, B coais a leas c1 γ veices, each sedig a edge of some colou, say j, o A. Le D B be hese veices. Now, ecallig ha c 2 ad 3, we have a coeced subgaph usig colous 1 ad j o a leas A D c1 γ c1 γ γ + 1 veices. So alogehe, he heoem holds if we iiially chose δ = δγ, such ha { 1 0 < δ mi 9, γ 4γ1 γ, , 16γ 2 1 γ 2 } γ 2. Now, Theoems 11 ad 12, ogehe wih Theoem 6 ad Lemma 7, imply immediaely Theoems 1 ad 2. Poof of Theoem 1. Give γ 0, 1 4 ad iege 3, le ε = γ ad 0 = 1 ε. Obai N 0 = N 0 ε,, 0 = N 0 γ, ad T 0 = T 0 ε,, 0 = T 0 γ, fom Theoem 6. Now, give K wih N 0, ad a -colouig f : EK [], le G i be he gaph o V K wih he edges of colou i, fo i []. The, hee exiss a paiio V K = V 0 V 1 V, fo some 1 ε T 0, which is simulaeously ε-egula wih espec o evey G i. We have V 0 ε, ad V i = V j fo all i, j 1. Now, fo each G i, we obai he cluse gaph R i 1 of he gaph G i V 0, so ha V R i 1 =. Le he gaph R 1 = i=1 R i 1, ad le g : ER 1 [] be a -colouig saisfyig guv {i [] : uv ER i 1 }, fo evey uv ER 1. Now, oe ha er 1 2 γ 2, sice hee ae a leas 2 ε 2 = 2 γ 2 pais V p, V q which ae ε-egula i evey G i, ad fo such a pai V p, V q, we have dv p, V q 1 i some G i. So, Theoem 11 implies ha we ca fid a moochomaic coeced subgaph of R 1 i he colouig g, say of colou l, o a leas 1 9γ 2 1 veices. This is a coeced subgaph of R l 1, so Lemma 7 implies ha G l has a [ 1 3γ V0 ] -coeced subgaph of ode a leas 1 γ 1 9γ V γ 1. The fis pa follows, sice 1 3γ V 0 1 4γ T 0. To see he secod pa, simply le γ 0 ad. The, 1 4γ T 0 equaliy pa comes fom he fac Lemma 8 of [8] ha m,, 1, k k+1 powe. 7 = o ad 6γ 1 1 = o. The + if 1 is a pime

8 Poof of Theoem 2. This is esseially he same as he pevious poof. We will use Theoem 12 isead of Theoem 11. Give γ 0, 1 4 ad iege 3, we fis obai δ = δγ, as give by Theoem 12, ad he se ε = δ ad 0 = 1 ε. Now, apply Theoems 6, 12, ad Lemma 7 as befoe. We obai he coespodig N 0 ad T 0, ad he akig a -colouig of EK, whee N 0, we obai he coespodig. This ime, we ge a 2-coloued, 1 4δ T 0 -coeced subgaph of K o a leas 1 δ δ ε δ γ + 1 veices. We ae doe, sice his subgaph is also 1 4γ T 0 -coeced. The secod pa is ow ivial, agai by leig γ 0 ad. The equaliy pa comes fom he fac Lemma 13 of [9] ha m,, 2, k if + 1 is a powe of Bipaie Regulaiy Lemmas ad Poof of Theoem 3 Havig ow poved Theoems 1 ad 2 wih he help of Szemeédi s Regulaiy Lemma, we would like o apply a simila idea o pove Theoem 3. We begi by discussig egulaiy lemma esuls coceig bipaie gaphs. Fisly, we will eed he followig vesio of he egulaiy lemma fo bipaie gaphs fo may colous. Theoem 13. Fo evey ε 0, 1 ad, 0 N, hee exis ieges N 0 = N 0 ε,, 0 ad T 0 = T 0 ε,, 0 such ha he followig holds. Evey bipaie gaph G = U V, E wih U =, V =, ad N 0, whose edges ae -coloued: E = E 1 E, admis a paiio of is veex se: U = U 0 U 1 U ad V = V 0 V 1 V, fo some 0 T 0, so ha U 0 ε ad V 0 ε, U i = U j ad V i = V j, fo evey 1 i, j, i, j, ad i evey subgaph G k = U V, E k, k [], all bu a mos ε 2 pais of {U i, V j } i,j 1 ae ε-egula. The poof of Theoem 13 is a saighfowad geealisaio see fo example [6] fo deails of he poof of he egulaiy lemma fo bipaie gaphs. Fo he poof of he lae see [12], see also [11] fo a weake fom ad [13] fo a moe ece poof as well. Ufouaely, if we aemp o use Theoem 13 diecly o ackle Theoem 3, we will u io a majo difficuly. Give a lage bipaie gaph G wih pa-sizes ad, whee, i us ou ha, whe we paiio V G, we would like all he cluse sizes o be oughly he same. If we use he paiio give by Theoem 13, his is ceaily fa fom beig ue if. So, ou ex aim is o suiably efie he paiio of V G as give by Theoem 13 while, i some sese, pesevig a lage popoio of ε-egula pais. We shall oughly divide ou cosideaio io he cases ad. Moe pecisely, give ε > 0, we coside he cases 33ε 5 ad > 33ε 5. Lemma 14 May-colous egulaiy lemma fo bipaie gaphs. Fo evey ε 0, 1 34 ad, 0 N, hee exis ieges N 0 = N 0 ε,, 0 ad T 0 = T 0 ε,, 0, such ha he followig holds. Le G = U V, E be a bipaie gaph, wih U =, V =, ad N 0. The, wheeve he edges of G ae -coloued: E = E 1 E, G admis a paiio of is veex se: U = U 0 U 1 U ad V = V 0 V 1 V, fo some 0 T 0, so ha U 0 ε ad V 0 ε, U i = V j fo all 1 i, 1 j, ad eihe, i evey G k = U V, E k, k [], all bu a mos ε pais of {U i, V j } i,j 1 ae ε-egula, if 33ε 5, o, all bu a mos ε pais of {U i, V j } i,j 1 ae ε, ε5 -egula, each oe wih espec o some colou, if > 33ε 5, < 4ε 5, ad G = K,. 8

9 The case > 33ε 5 will be he ickie case. We shall deive a key lemma o help us o pove his case. To do his, we fis ecall a lemma of Alo e al. [1], which has a sufficie codiio fo a bipaie gaph o be ε-egula. Fo a pai V 1, V 2 wih V 1 = V 2 =, desiy d, ad Y V 1, defie he deviaio of Y by whee fo disic y 1, y 2 V 1 is he eighbouhood deviaio of y 1 ad y 2. Hee he, is he lemma of Alo e al. σy = 1 Y 2 y 1,y 2 Y σy 1, y 2, σy 1, y 2 = Γy 1 Γy 2 d 2 Lemma 15 Regulaiy cieio; Lemma 3.2 of [1]. Le G be a bipaie gaph wih classes V 1 ad V 2, whee V 1 = V 2 =, ad desiy d. Le 2 1/4 < ε < Assume ha {x V 1 : degx d ε 4 } 1 8 ε4, ad ha fo evey Y V 1 wih Y ε, we have σy < 1 2 ε3. The, G is ε-egula. Now, hee is he lemma ha we will equie. Lemma 16. Le ε 0, 1 16, k N ad X 1, X,..., X k, X be 1 16 ε5 -egula pais, wih d i := dx i, X 1 4 ε5, 1 fo evey i, ad moeove, X i = m fo evey i, X = m, m m > 16ε 4, ad lm = m fo some l N. The, hee exiss a paiio X = U 1 U l, whee U j = m fo evey j, such ha a leas [ 1 2m 3 exp 3 ] 16 4 ε20 m kl of he pais X i, U j ae ε-egula, ad dx i, U j d i 1 4 ε5, d i ε5. Poof. We shall pove ha, by akig a adom paiio of X io pas of size m, he coclusio holds wih pobabiliy a leas 1 1 m > 0. To do his, we shall apply Lemma 15. We show ha fo evey i, mos veices fom X i ad mos pais of veices fom X i have oughly he expeced degees ad co-degees i a adomly chose subse of X of size m. Fix X i, ad le {v 1,..., v m } be is veex se. Fo evey 1 m, le Z be he adom vaiable ha cous he eighbous of he veex v X i i a subse U X of size m, chose uifomly a adom. Z has a hypegeomeic disibuio Hgm, m, degv. Le ε = 1 16 ε5. By Lemma 4, all bu a mos 2 εm veices of X i have degees i X lyig i he ieval d i m εm, d i m + εm. Le X i be hese veices, ad v X i. By Cheoff s iequaliy see, fo example, Theoem 2.10 i [5], we have PZ d i m 2 εm P Z degv εm m m P Z 1 ε degv m m exp ε2 degv m 3m exp 1 3 ε2 d i εm, ad similaly, PZ d i m+2 εm exp 1 3 ε2 d i εm. Applyig his o evey such veex i k i=1 X i, we have P Γv U d i m 2 εm, d i m + 2 εm wheeve v 1 k i=1 k X i, v X i i=1 2m exp 1 3 ε2 d i εm 1 2km exp ε15 m. Nex, fo fixed X i ad fo evey odeed pai v s, v of veices fom X i, le he adom vaiable Z s be he umbe of commo eighbous of v s ad v i a adomly chose subse U of X of size m. We have Z s Hg m, m, Γv s Γv. Le X i X i X i be hose odeed pais wih commo degee 9

10 i X lyig i d i ε 2 m, d i + ε 2 m. By Lemma 4, X i cosiss of all bu a mos 4 εm 2 pais fom X i. Agai, by Cheoff s iequaliy, fo v s, v X i, P Z s d i + ε 2 m + εm exp 1 3 ε2 d i ε 2 m, by a simila calculaio. Applyig his o all such odeed pais i k P Γv s Γv U < d i + ε 2 m + εm wheeve v s, v 1 k i=1 k i=1 m 2 exp 1 3 ε2 d i ε 2 m i=1 X i, we have X i, v s, v X i 1 km 2 exp ε20 m. So ow, choose a se U X wih U = m, uifomly a adom. Fo each 1 i k, le G i = X i, U ad d i = dx i, U. Wih pobabiliy a leas 1 2km 2 exp ε 20 m, we have, fo evey i, ad, d i d im 2 εmm 2 εm m 2 d i 4 ε = d i 1 4 ε5, d i d im + 2 εmm 2 εm + 2 εm m m 2 d i + 4 ε = d i ε5. Wih hese, i is ow easy o show ha fo evey i, {x Xi : deg Gi x d im ε 4 m} {x Xi : deg Gi x d i m 2 εm} 2 εm 1 8 ε4 m. Nex, fix i, ad le Y X i wih Y εm. Le Y 1 be hose odeed pais of Y which ae i X i, ad Y 2 be hose which ae o. So, Y 2 4 εm 2. We have σy = 1 Y 2 σy 1, y 2 + σy 1, y 2 y 1,y 2 Y 1 y 1,y 2 Y 2 1 Y 2 [ di + ε 2 m + εm d i 4 ε 2 m Y 2 Y 2 + m d i 4 ε 2 m Y 2 ] = 1 Y 2 [ di + ε 2 + ε d i 4 ε 2 m Y d i + ε 2 ε m Y 2 ] < 11 εm + 4 εm ε 2 = ε5 m ε3 m < 1 2 ε3 m. Sice 2m 1/4 < ε < 1 16, Lemma 15 implies ha X i, U is ε-egula. This is he ue fo evey i, wih pobabiliy a leas 1 2km 2 exp 3 16 ε 20 m. 4 Fially, fo a adom paiio X = U 1 U l, whee U j = m fo evey j, le he adom vaiable N be he umbe of pais X i, U j which ae o boh ε-egula ad wih dx i, U j d i 1 4 ε5, d i ε5. The, EN kl 2m 2 exp 3 16 ε 20 m. Thus, by Makov s iequaliy, we have 4 P N kl 2m 3 exp ε20 m 1 m, which implies he lemma. We ae ow eady o pove Lemma 14. Poof of Lemma 14. Le such ε, ad 0 be give. Le G = U V, E be a bipaie gaph wih U =, V = ad, ad whose edges ae -coloued: E = E 1 E. We coside wo cases. Case 1 33ε 5. Le N 0 ad T 0 be he ieges obaied fom Theoem 13, usig ε = ε 8. Choose N 0 = N 0 ad T 0 T 0ε 7. If N 0, he Theoem 13 applies fo G. So, we have a paiio U = U 0 U 1 U s ad V = V 0 V 1 V s, fo some 0 s T 0, so ha U 0 ε 8 ad V 0 ε 8, 10

11 U i = m ad V j = m fo all 1 i, j s ad some m, m, ad i evey G k = U V, E k, k [], all bu a mos ε 8 s 2 pais of {U i, V j } i,j 1 ae ε 8 -egula. Fo each U i, divide i io subses of size ε7 m, leavig a emaiig se of size less ha ε 7 m. Uie hese emaiig ses wih U 0 ad le U 0 be he uio. The, U 0 ε 8 + ε 7 m m ε8 + 33ε2 1 ε ε 8 if ε < Repea his wih each V j, agai dividig io ses of size ε7 m, ad le V 0 be he aalogous uio of V 0 wih he emaiig ses. The, V 0 ε 8 + ε 7 ε. Now, le U 1,..., U U ad V 1,..., V V be he subses of size ε 7 m i he above paiio. If U p U i ad V q V j, ad U i, V j is ε8 -egula i evey colou, he applyig Lemma 5 wih α = ε 7 gives ha U p, V q is ε-egula i evey colou. So, hee ae a leas 1 ε 8 s 2 m ε 7 m m ε 7 m = 1 ε8 1 ε such pais U p, V q. Fially, = m ε 7 m s T 0, so 0 s T 0. This poves Case 1. Case 2 > 33ε 5, < 4ε 5 ad G = K,. Le N 0 ad T 0 be he ieges obaied fom Theoem 13, usig ε = 1 33 ε5. Choose N 0 > max 16ε 4 T 0 1 ε, θt 0 1 ε, N 0 ad T0 = T 0, whee θ = θε is he lages soluio o 2θ 3 exp 3 16 ε 20 θ = ε. If N 0, he Theoem 13 applies fo K,. So, we have a paiio U = U 0 U 1 U ad V = W 0 W 1 W, whee 0 T 0, such ha U ε5 ad W ε5, U i = m ad W j = m, fo evey 1 i, j ad some m, m, ad i evey G k = U V, E k, k [], all bu a mos 1 33 ε5 2 pais of {U i, W j } i,j 1 ae 1 33 ε5 -egula. We have U 0 ε. Now, le m = lm+l, whee l, l Z ad 0 l < m. Fo each W j, emove a se of size l ad uie hese ses wih W 0, fomig a ew se V 0. The, V ε ε5 W 0 ε. Also, oe ca show ha l > 33ε 5 2, so ha l < m l+1 < 1 32 ε5 m. Le W j be he emaiig se fom W j Wj. Now, W j l = 1 m > ε5. If U i, W j is 1 33 ε5 -egula i evey colou, he applyig Lemma 5 wih α = ε5 gives ha U i, W j is 1 16 ε5 -egula i evey colou. Now, coside he subgaph H K, as follows. V H = V K, \ U 0 V 0. If U i, W j is 1 16 ε5 -egula i evey colou, choose a colou whose desiy is a leas 1, ad keep oly hese edges fo H. Fially, disegad he colous. Now, i H, fo fixed W j, le he pais U j 1, W j,..., U j k j, W j be pecisely he 1 16 ε5 -egula pais, whee U j 1,..., U j k j {U 1,..., U }, ad wih desiies a leas 1, so ha du j 1, W j,..., du j k j, W j 1 4 ε5, 1. By he choices of N 0 ad T 0, Lemma 16 applies o he pais U j 1, W j,..., U j k j, W j. So, hee is a paiio W j = V j 1 V j l io ses of size m, such ha a leas [ 1 2m 3 exp 3 16 ε 20 m ] k 4 j l of he pais Up, j Vq j ae ε-egula, wih desiies geae ha ε5. Now, akig he uio ove W j ad oig ha m > θ, a leas j=1 [ 1 2m 3 exp 3 ] 16 4 ε20 m k j l ε ε5 2 l 1 ε pais U p, V q ae ε-egula, wih desiies geae ha ε5, whee {V 1,..., V } ae all he V j q. Case 2 follows afe we eisae he colous of he edges fom H. This complees he poof of Lemma 14. Havig obaied he bipaie egulaiy lemma ha we will equie: Lemma 14, we ca ow poceed o pove Theoem 3. Fisly, we have he followig aalogous saeme o Theoems 11 ad 12 fo bipaie gaphs, which is a easy coollay of Lemma 10. Lemma 17. Le γ 0, 1, ad m,, N. Le G be a bipaie gaph wih pa-sizes m ad, ad eg m γm. The, if we have a -colouig of EG, hee is a moochomaic coeced subgaph o a leas m+ γm+ veices. 11

12 To pove Theoem 3, we use a simila idea as i he ed of Secio 3. Hee, we shall use Lemmas 14, 17 ad 7. Poof of Theoem 3. We fis pove he lowe boud. Le γ 0, 1 34 ad,, N, whee 2 ad. Le ε = γ ad 0 = 1 ε. Obai N 0 ad T 0 fom Lemma 14. Now, give K, = U V, E wih U = N 0 ad V =, ad a -colouig of i, we have a paiio U = U 0 U 1 U ad V = V 0 V 1 V such ha U 0 ε, V 0 ε, U i = V j = m fo all i, j 1 ad some m, ad 1 ε T 0. Now, we coside wo cases, defiig ou cluse gaphs slighly diffeely i each case. Case 1 33ε 5. I his case, a leas 1 ε pais of {U i, V j } i,j 1 ae ε-egula i evey colou. By cosideig cluse gaphs wih η = 1 of he coloued subgaphs o K, U 0 V 0, i follows ha, by Lemmas 17 ad 7, we have a 1 4γ T 0 -coeced, moochomaic subgaph o a leas 1 γ + γ + + ε ε + + 3γ + veices, sice m = U0 = V 0, so ha m = + U 0 V ε ε +. Case 2 > 33ε 5. I his case, a leas 1 ε pais of {U i, V j } i,j 1 ae ε, ε5 -egula, each oe i some colou. Fo each i [], le G i be he subgaph of K,, wih he edges of colou i. Le R i ε5 be he cluse gaph deived fom G i U 0 V 0, ad R ε5 = i=1 R i ε5. The, e R ε5 1 ε 1 γ. As befoe, Lemmas 17 ad 7 imply ha some G l has a [ ε5 3εm]-coeced subgaph o a leas + 3γ+ veices, by he same calculaio as i Case 1. Case 2 follows, sice ε5 3εm 1 4γ T 0 if 2. Now, we show he uppe boud. Fo k N, we shall descibe a -colouig of EK,, whee k, such ha he lages moochomaic k-coeced subgaph has ode a mos So, ake such a K, = U V, E, whee U = ad V =. Paiio each of U ad V io pas: U = U 1 U, ad V = V 1 V, each as equally as possible. Noe ha U i, V j k fo evey i, j. Now, colou he edges of EU i, V j wih colou i j mod. The, he lages moochomaic k-coeced subgaph has ode The fial pa is ow ivial, agai by leig γ 0 ad, ad usig a simila agume as befoe. 5. Ope poblems Havig ow impoved he lowe bouds fo m,, 1, k ad m,, 2, k fo lage, a obvious quesio o ask would be wha happes fo s 3. Ou mai poblem is ha we do o have aalogous esuls o Theoems 11 ad 12 fo s 3. Noe ha fo some special values of Theoem 11 says ha mg,, 1, 1 is close o m,, 1, 1 if G is close o K, ad Theoem 12 says a simila hig fo s = 2. So, we pose he followig cojecue, which asks fo a aalogous saeme fo s 3, as well as fo hose values of whee we do o kow he value of m,, s, 1, fo s = 1, 2. Cojecue 18. Le γ 0, 1, ad,, s N wih s, 3 ad s 1. The hee exiss δ = δγ,, s wih 0 < δ < γ such ha he followig holds. If G is a gaph o veices wih eg 2 δ 2, he mg,, s, 1 1 γm,, s, 1. If some fom of Cojecue 18 holds, he we ca coceivably apply he same echiques, ad ge m,, s, k m,, s, 1 o fo, s fixed, 3 ad s 1, ad k = o. Thus, we may oly have o woy abou 1-coecedess if we wee o ackle Bollobás quesio i his diecio. Aohe poblem is ha we have o said much abou m bip,,, s, k fo s 2. I is easy o adap he poof of Theoem 3 ad ge m bip,,, s, k s+ o. Bu his could be weak excep fo s = 2, whee we may slighly adjus he uppe boud cosucio i he poof of Theoem 3, ad ge ha m bip,,, 2, k if is a powe of 2 his is simila o he cosucio give i Lemma 13 of [9]. So, we fiish by saig he poblem iself. Poblem 19. Deemie m bip,,, s, k fo s 2. 12

13 6. Ackowledgemes The auhos would like o hak Mahias Schach fo his useful discussios o he poof of Lemma 16. The fis auho would also like o hak Humbold-Uivesiä zu Beli fo hei geeous hospialiy. He was able o cay ou pa of his eseach wih he secod auho duig his visi hee. Refeeces [1] N. Alo, R. A. Duke, H. Lefma, V. Rödl ad R. Yuse, The algoihmic aspecs of he egulaiy lemma, J. Algoihms, , [2] B. Bollobás, pesoal commuicaio. [3] B. Bollobás, Mode Gaph Theoy, Spige-Velag, New Yok, 1998, xiii+394pp. [4] B. Bollobás ad A. Gyáfás, Highly coeced moochomaic subgaphs, Discee Mah., , [5] S. Jaso, T. Luczak ad A. Ruciński, Radom gaphs, Wiley-Iesciece Seies i Discee Mahemaics ad Opimizaio, New Yok, 2000, xii+333pp. [6] J. Komlós ad M. Simoovis, Szemeédi s egulaiy lemma ad is applicaios i gaph heoy, Paul Edős is eighy, Poceedigs of Colloquia of he Bolyai Mahemaical Sociey 2 Keszhely, 1993, [7] H. Liu, R. Mois ad N. Pice, Highly coeced moochomaic subgaphs: addedum, mauscip. [8] H. Liu, R. Mois ad N. Pice, Highly coeced moochomaic subgaphs of mulicoloued gaphs, J. Gaph Theoy, , [9] H. Liu, R. Mois ad N. Pice, Highly coeced mulicoloued subgaphs of mulicoloued gaphs, Discee Mah., , [10] W. Made, Exisez -fach zusammehägede Teilgaphe i Gaphe geüged gosse Kaediche, Abh. Mah. Sem. Uiv. Hambug, , [11] E. Szemeédi, O ses of ieges coaiig o k elemes i aihmeic pogessio, Aca Aihmeica , [12] E. Szemeédi, Regula paiios of gaphs, Poblèmes combiaoies e héoie des gaphes, Poceedigs du Colloque Ieaioal CNRS, 260, CNRS, Pais, 1978, [13] T. Tao, Szemeédi s egulaiy lemma evisied, Coib. Discee Mah., ,