The Erdős-Gyárfás problem on generalized Ramsey numbers

Transcription

1 The rdős-gyárfás problem on generalzed Ramsey numbers Davd Conlon Jacob Fox Choongbum Lee Benny Sudakov Abstract Fx postve ntegers p and q wth 2 q p 2. An edge-colorng of the complete graph Kn s sad to be a p, q-colorng f every K p receves at least q dfferent colors. The functon fn, p, q s the mnmum number of colors that are needed for K n to have a p, q-colorng. Ths functon was ntroduced by rdős and Shelah about 40 years ago, but rdős and Gyárfás were the frst to study the functon n a systematc way. They proved that fn, p, p s polynomal n n and asked to determne the maxmum q, dependng on p, for whch fn, p, q s subpolynomal n n. We prove that the answer s p 1. 1 ntroducton The Ramsey number r k p s the smallest natural number n such that every k-colorng of the edges of the complete graph K n contans a monochromatc K p. The exstence of r k 3 was frst shown by Schur [13] n 1916 n hs work on Fermat s Last Theorem and t s known that r k 3 s at least exponental n k and at most a multple of k!. t s a central problem n graph Ramsey theory to close the gap between the lower and upper bound, wth connectons to varous problems n combnatorcs, geometry, number theory, theoretcal computer scence and nformaton theory see, e.g., [9, 10]. The followng natural generalzaton of the Ramsey functon was frst ntroduced by rdős and Shelah [3, 4] and studed n depth by rdős and Gyárfás [5]. Let p and q be postve ntegers wth 2 q p 2. An edge-colorng of the complete graph Kn s sad to be a p, q-colorng f every K p receves at least q dfferent colors. The functon fn, p, q s the mnmum number of colors that are needed for K n to have a p, q-colorng. Mathematcal nsttute, Oxford OX2 6GG, Unted Kngdom. mal: davd.conlon@maths.ox.ac.uk. Research supported by a Royal Socety Unversty Research Fellowshp. Department of Mathematcs, MT, Cambrdge, MA mal: fox@math.mt.edu. Research supported by a Packard Fellowshp, by a Smons Fellowshp, by NSF grant DMS , by an Alfred P. Sloan Fellowshp and by an MT NC Corporaton Award. Department of Mathematcs, MT, Cambrdge, MA mal: cb lee@math.mt.edu. Department of Mathematcs, TH, 8092 Zurch, Swtzerland and Department of Mathematcs, UCLA, Los Angeles, CA mal: benjamn.sudakov@math.ethz.ch. Research supported n part by SNSF grant and by a USA-srael BSF grant. 1

2 To see that ths s ndeed a generalzaton of the usual Ramsey functon, note that fn, p, 2 s the mnmum number of colors needed to guarantee that no K p s monochromatc. That s, fn, p, 2 s the nverse of the Ramsey functon r k p and so we have c log n log log n fn, 3, 2 c log n. rdős and Gyárfás [5] proved a number of nterestng results about the functon fn, p, q, demonstratng how the functon falls off from beng equal to n 2 when q = p 2 to beng at most logarthmc when q = 2. n so dong, they determned ranges of p and q where the functon fn, p, q s lnear n n, where t s quadratc n n and where t s asymptotcally equal to n 2. Many of these results were subsequently sharpened by Sárközy and Selkow [11, 12]. One smple observaton made by rdős and Gyárfás s that fn, p, p s always polynomal n n. To see ths, t s suffcent to note that f a colorng uses fewer than n 1/p 2 1 colors then t necessarly contans a K p whch uses at most p 1 colors. For p = 3, ths s easy to see snce one only needs that some vertex has at least two neghbors n the same color. For p = 4, we have that any vertex wll have at least n 1/2 neghbors n some fxed color. But, snce there are fewer than n 1/2 1 colors on ths neghborhood of sze at least n 1/2, the case p = 3 mples that t contans a trangle wth at most two colors. The general case follows smlarly. rdős and Gyárfás [5] asked whether ths result s best possble, that s, whether q = p s the smallest value of q for whch fn, p, q s polynomal n n. For p = 3, ths s certanly true, snce we know that fn, 3, 2 c log n. However, for general p, they were only able to show that fn, p, log p s subpolynomal, where here and throughout the paper we use log to denote the logarthm taken base 2. Ths left the queston of determnng whether fn, p, p 1 s subpolynomal wde open, even for p = 4. The frst progress on ths queston was made by Mubay [8], who found an elegant constructon whch mples that fn, 4, 3 e c log n. Ths constructon was also used by chhorn and Mubay [2] to demonstrate that fn, 5, 4 e c log n. More generally, they used the constructon to show that fn, p, 2 log p 2 s subpolynomal for all p 5. n ths paper, we answer the queston of rdős and Gyárfás n the postve for all p. That s, we prove that fn, p, p 1 s subpolynomal for all p. Quanttatvely, our man theorem s the followng. Theorem 1.1. For all natural numbers p 4 and n 1, fn, p, p plog n1 1/p 2 log log n. n Secton 2, we defne our p, p 1-colorng by a recursve procedure. We begn by revewng Mubay s 4, 3-colorng, as t s the base case of our recurson. The formal proof of the fact that our colorng s ndeed a p, p 1-colorng s qute techncal and thus we frst gve an outlne of the proof n Secton 3. Then, n Secton 4, we establsh some propertes of the colorng. Fnally, n 2

3 Secton 5, we prove that the colorng gven n Secton 2 s a p, p 1-colorng. We wll conclude wth some further remarks. Notaton. For vectors v X t 1+t 2, v 1 X t 1, v 2 X t 2, we wll often use the notaton v = v 1, v 2, n order to ndcate that the -th coordnate of v s equal to the -th coordnate of v 1 for 1 t 1 and the t 1 + j-th coordnate of v s equal to the j-th coordnate of v 2 for 1 j t 2. We wll use smlar notaton for several vectors. Throughout the paper, log denotes the base 2 logarthm. For the sake of clarty of presentaton, we systematcally omt floor and celng sgns whenever they are not essental. 2 The colorng constructon The purpose of ths secton s to defne the colorng used to prove Theorem 1.1. The colorng can be consdered as a generalzaton of a varant of Mubay s 4, 3-colorng. We therefore frst ntroduce ths colorng and then redefne t n a way that can be naturally extended. We then present the colorng used to prove Theorem 1.1. As t s a rather nvolved recursve defnton, we gve an example to llustrate t. We conclude the secton by establshng a bound on the number of colors used n ths colorng. n the followng sectons, we wll show that ths colorng s a p, p 1-colorng, completng the proof. 2.1 Mubay s 4, 3-colorng Let N = m t for some ntegers m and t. Suppose that we are gven two dstnct vectors v, w [m] t of the form v = v 1,..., v t and w = w 1,..., w t. Defne cv, w = {v, w }, a 1,..., a t, where s the least coordnate n whch v w and a j = 0 f v j = w j and a j = 1 f v j w j. f v = w, defne cv, v = 0. Note that c s a symmetrc functon. Ths s a varant of Mubay s colorng and can be proved to be a p, p 1-colorng for small values of p. One mght suspect that ths s a p, p 1-colorng for large ntegers p as well, but, unfortunately, t fals to be a 26, 25-colorng and a p, p 1-colorng for all p 26 for the followng reason. Consder the set {1, 2, 3} 3. Ths set has 3 3 = 27 elements and at most = 24 colors are used n colorng ths set. Therefore, we can fnd 26 vertces wth at most 24 colors wthn the set. Moreover, for every fxed p and large enough N, lettng s = log p, the set S = {1, 2,..., 2 s } s has cardnalty 2 s2 = p and uses at most 2 s 2 2 s < 2 3s = 2 3 log p colors and, for large enough m and t, s a subset of [m] t. Hence, ths edge-colorng of the complete graph on [N] fals to be a p, 2 3 log p -colorng. 3

4 2.2 Redefnng Mubay s colorng Before proceedng further, let us redefne the colorng gven above from a slghtly dfferent perspectve. We do ths to motvate the p, p 1-colorng whch we use to establsh Theorem 1.1. Let m = 2 r 1 and, abusng notaton, dentfy the set [m] wth {0, 1} r 1. Let r 2 = r 1 t for some postve nteger t. Suppose that we are gven two vectors v, w [m] t = {0, 1} r1t. We decompose v as v = v 1 1,..., v1 t, where v 1 {0, 1} r 1 for = 1, 2,..., t and smlarly decompose w. The functon c was defned as follows: cv, w = {v 1, w 1 }, a 1,..., a t, where s the least coordnate n whch v 1 w 1 and, for j = 1, 2,..., t, a j represents whether v 1 j = w 1 j or not. f v = w, then cv, v = 0. Defne h 1 as the frst coordnate of c. That s, h 1 v, w = {v 1, w 1 } we let h 1 v, v = 0 for convenence. Note that h 1 takes a par of vectors of length r 2 = r 1 t as nput and outputs a par of vectors of length r 1. For two vectors x, y {0, 1} r 1 of the form x = x 1,..., x r1, y = y 1,..., y r1, defne the functon h 0 as follows. We have h 0 x, x = 0 for each x and, f x y, then h 0 x, y = {x, y }, where s the mnmum ndex for whch x y. Snce all x and y are ether 0 or 1, there are only two possble outcomes for h 0, 0 f the two vectors are equal and {0, 1} f they are not equal. Note that h 0 takes a par of vectors of length r 1 as nput and outputs a par of vectors of length r 0 = 1. Thus, both h 1 and h 0 are functons whch record the frst block that s dfferent. The dfference between the two functons les n ther nterpretaton of block : for h 1 t s a subvector of length r 1 and for h 0 t s a subvector of length r 0. Summarzng, we see that c s equvalent to the colorng c gven by c v, w = h 1 v, w, h 0 v 1 1, w1 1,..., h 0v 1 t, w 1 t. nformally, we frst decompose the gven par of vectors v and w nto subvectors of length r 2 and apply h 1 we observe only a sngle subvector n ths case snce v and w themselves are vectors of length r 2. Then we decompose v and w nto subvectors of length r 1 and apply h 0 to each correspondng par of subvectors of v and w. 2.3 Defnton of the colorng n ths secton, we generalze the constructon gven n the prevous secton to obtan a p, p 1- colorng. For a postve nteger α, we wll descrbe the colorng as an edge-colorng of the complete graph over the vertex set {0, 1} α. Let r 0, r 1,... be a sequence of postve ntegers such that r 0 = 1 and r d 1 dvdes r d for all d 1. 4

5 For a set of ndces, let π be the canoncal projecton map from {0, 1} α to {0, 1}. We wll wrte π nstead of π [] for convenence. Thus π s the projecton map to the frst coordnates. The key dea n the constructon s to understand vectors at several dfferent resolutons. Suppose that we are gven two vectors v, w {0, 1} α. For d 0, let a d and b d be ntegers satsfyng a d 0 and 1 b d r d such that α = a d r d + b d. Let v = v d 1, vd 2,..., vd a d +1 where v d {0, 1} r d for = 1, 2,..., a d and v d a d +1 {0, 1}b d. We refer to the vectors v d as blocks of resoluton d. We smlarly decompose w as w = w d 1, wd 2,..., wd a d, w d a d +1 for d 0. We frst defne two auxlary famles of functons η d and ξ d. For d 0, f v w, let η d v, w = where s the mnmum ndex such that v d, {v d, w d },, w d. f v = w, let η d v, v = 0. Note that η d s a symmetrc functon. Further note that η d s slghtly dfferent from h d defned n the prevous subsecton snce we add an addtonal coordnate whch records the ndex as well. The man theorem s vald even f we do not add ths ndex, but we choose to add t as t smplfes the proof. We refer the reader to Subsecton 6.2 for a further dscusson of ths pont. For d 0, let ξ d v, w = d+1 η d v 1, w d+1 d+1 1,..., ηd v a d+1 +1, wd+1 a d Note that the functon ξ d decomposes the vectors nto blocks of resoluton d + 1 and outputs a vector contanng nformaton about blocks of resoluton d. For d 0, let c d = ξ d ξ d 1... ξ 0. Note that the colorng c d depends on the choce of the parameters r 0, r 1,..., r d+1. We prove our man theorem n two steps: we frst estmate the number of colors and then prove that t s a p, p 1-colorng. Theorem 2.1. Let p and β be fxed postve ntegers wth β 1. For the choce r = β for 0 p + 1, the edge-colorng c p of the complete graph on n = 2 βp+1 vertces uses at most 2 4log n1 1/p+1 log log n colors. Theorem 2.2. Let p and α be fxed postve ntegers. Then, for every choce of parameters r 1,..., r p+1, the edge-colorng c p s a p + 3, p + 2-colorng of the complete graph on the vertex set {0, 1} α. 5

6 For ntegers n of the form n = 2 βp+1, Theorem 1.1 follows from Theorems 2.1 and 2.2. For general n p + 3 4, frst notce that f n 2 < 2 16plog n1 1/p+1 log log n, then the statement s trvally true, as we may color each edge wth dfferent colors. Hence, we may assume that the nequalty does not hold, from whch t follows that 2 log n 16plog n 1 1/p+1 log log n 16plog n 1 1/p+1 and n 2 8pp+1. Hence, there exsts an nteger of the form 2 βp+1 whch s at most n 1+1/8pp+1 n 2. Therefore, there exsts a p + 3, p + 2-colorng of the complete graph on the vertex set [n] usng at most 2 42 log n1 1/p+1 log2 log n 2 4 2log n1 1/p+1 1+log log n 2 16log n1 1/p+1 log log n colors n the second nequalty we used the fact that log log n log log 4 1. Thus we obtan Theorem 1.1. Theorem 2.1 s proved n Subsecton 2.5, whle Theorem 2.2 s proved n Secton 5 and bulds on the two sectons leadng up to t. 2.4 xample Let us llustrate the colorng by workng out a small example. Suppose that r 1 = 2 and r 2 = 4. Let v = 0, 0, 1, 0, 1, 1, 0 and w = 0, 0, 1, 1, 1, 0, 0 be vectors n {0, 1} 7. Then v = 0, 0, 1, 0, 1, 1, 0 = 0, 0, 1, 0, 1, 1, 0 = 0, 0, 1, 0, 1, 1, 0, where the quotaton marks ndcate the blocks of each resoluton. Smlarly, w = 0, 0, 1, 1, 1, 0, 0 = 0, 0, 1, 1, 1, 0, 0 = 0, 0, 1, 1, 1, 0, 0. The functon η 0 records the frst par of blocks of resoluton 0 whch are dfferent. So η 0 v, w = 4, {0, 1}, where the value of the frst coordnate, 4, ndcates that v and w frst dffer n the fourth coordnate. Smlarly, the functon η 1 wll record the frst par of blocks of resoluton 1 whch are dfferent. So η 1 v, w = 2, {1, 0, 1, 1}. Computng ξ 0 and ξ 1 nvolves one more step. To compute ξ 0, we apply η 0 to each par of blocks of resoluton 1. Therefore, ξ 0 v, w = η 0 0, 0, 0, 0, η0 1, 0, 1, 1, η0 1, 1, 1, 0, η0 0, 0 whch s a vector of length four. = 0, 2, {0, 1}, 2, {1, 0}, 0, 6

7 Smlarly, to compute ξ 1, we apply η 1 to each par of blocks of resoluton 2. Therefore, ξ 1 v, w = η 1 0, 0, 1, 0, 0, 0, 1, 1, η1 1, 1, 0, 1, 0, 0 = 2, { 1, 0, 1, 1 }, 1, { 1, 1, 1, 0 }, whch s a vector of length two. 2.5 Number of colors n ths subsecton, we prove Theorem 2.1. Proof of Theorem 2.1. Recall that β s a postve nteger greater than 1 and r d = β d for 0 d p + 1. Let α = β p+1. The goal here s to gve an upper bound on the number of colors n the edge-colorng c p of the complete graph wth vertex set {0, 1} α = {0, 1} βp+1. Frst, for 0 d p, the functon η d outputs ether zero or an ndex and a par of dstnct blocks of resoluton d. Hence, there are at most 1 + α 2 r d2 r d 1 α2 2βd possble outcomes for the functon η d. Second, for α 0 d p, the functon ξ d s a product of r d+1 = β p d outcomes of η d. Hence, there are at most α 2 2β d β p d = β p+1β p d 2 2βp possble outcomes for the functon ξ d. Snce c p s defned as ξ p ξ p 1 ξ 0, the total number of colors used n c p s at most p d=0 β p+1βp d 2 2βp β 2p+1βp 2 2p+1βp 2 4p+1βp log β. Let n = 2 α = 2 βp+1 and note that β p = log n 1 1/p+1 and log β = 1 p+1 log log n. Thus, we have colored the edges of the complete graph on n vertces usng at most colors, as clamed n Theorem log n1 1/p+1 log log n As we saw n Subsecton 2.1, for large enough q, Mubay s colorng whch s smlar to c 1 s not a q, q 1-colorng or even a q, q ε -colorng for any fxed ε > 0. Smlarly, we can see that the same s true for the colorng c p for every fxed p we wll brefly descrbe the proof of ths fact n Subsecton 6.3. Ths explans why we need to consder c p wth an ncreasng value of p. 7

8 3 Outlne of proof n ths secton, we outlne the proof of Theorem 2.2. Assume that we want to prove that the edge-colorng of the complete graph on the vertex set {0, 1} α gven by c p s a p + 3, p + 2-colorng. We wll use nducton on α to prove the stronger statement that the colorng s a q, q 1 colorng for all q p + 3. To llustrate a smple case, assume that we are about to prove t for α = r p+1 and have proved t for all smaller values of α. Let S {0, 1} α be a gven set of sze at most p + 3. We wsh to show that the edges of S receve at least S 1 dstnct colors. Let α = r p+1 r p. For two vectors v, w S satsfyng v w, let v = v, v and w = w, w where v, w {0, 1} α and v, w {0, 1} α α = {0, 1} rp. Note that snce α = r p+1 r p s dvsble by r p, the frst α r p blocks of resoluton p of v are dentcal to those of v and a smlar fact holds for w and w. f v = w then, by the observaton above, the frst α r p coordnates of ξ p 1 are all zero. On the other hand, f v w, then the frst block of resoluton p on whch v and w dffer s one of the frst α r p blocks. Hence, n ths case, at least one of the frst α r p coordnates of ξ p 1 s non-zero. Thus, f we defne sets Λ and Λ as Λ = { c p v, w : v w, v, w S } and Λ = { c p v, w : v = w, v w, v, w S }, then we have Λ Λ =. Hence, t suffces to prove that Λ + Λ S 1. The ndex stands for nherted colors and stands for emergng colors. The colorng c p contans more nformaton than necessary to prove that the number of colors s large. Hence, we consder only part of the colorng c p. The part of the colorng that we consder for Λ and Λ wll be dfferent, as we would lke to hghlght dfferent aspects of our colorng dependng on the stuaton. Defne the sets C and C as { cp C = v, w, η p 1 v, w } : v w, v, w S and C = { } {v, w } : v = w, v w, v, w S. We clam here wthout proof that C Λ and C Λ. Abusng notaton, for two vectors v, w S, we wll from now on refer to the color between v and w as the correspondng color n C or C. t now suffces to prove that C + C S 1. To analyze the colors n C and C, we take a step back and consder the frst α coordnates of the vectors n S. Let S = π α S. Note that S s the collecton of vectors v n the notaton above. There s a certan branchng phenomenon of vectors and colors. For a vector v S, let 8

9 T v = {v : π α v = v, v S}. Hence, T v s the set of vectors n S whose frst α coordnates are equal to v. Note that T v = S. 1 v S Consder two vectors v, w S. f v and w are both n the same set T v, then the color between v and w belongs to C and f they are n dfferent sets, then the color between v and w belongs to C. For a color c C, note that the frst coordnate of c s of the form c p v, w for two vectors v, w S. Further note that c p v, w s the color of an edge that les wthn S. Hence, c s a branch of some color of an edge that les wthn S. n partcular, by nducton on α, we see that C S 1. 2 For a color c C, let µ c be the number of unordered pars of vectors v, w such that c s the color between v and w. We have the followng equaton Tv µ c = T v c C v S v S Let us frst consder the smple case when µ c = 1 for all c C that s, there are no overlaps between the emergng colors. n ths case, we have C = c C µ c. By 2, we have whch by 3 and 1 s at least C + C S 1 + C = S 1 + c C µ c, S 1 + T v 1 = T v 1 = S 1 v S v S and thus the concluson follows for the case when µ c = 1 for all c C. However, there mght be some overlap between the emergng colors. Note that there are C emergng colors nstead of the c C µ c whch we obtan by countng wth multplcty. Thus, there are c C µ c 1 lost emergng colors. Our key lemma asserts that every lost emergng color wll be accounted for by contrbutons towards C. Formally, we wll mprove 2 and obtan the followng nequalty C S 1 + µ c 1. 4 c C Gven ths nequalty, we wll have C + C S 1 + whch, as above, mples that C + C S 1. c C µ c 1 + C = S 1 + c C µ c, 9

10 We conclude ths secton wth a sketch of the proof of 4. To see ths, we further study the branchng of the colors. Defne C B as the set of colors that appear wthn the set S, that s, C B = { c p v, w : v, w S }, where the ndex B stands for base colors. very color c C s of the form c = c,?, where c C B and the queston mark? stands for an unspecfed coordnate. Thus, we mmedately have at least C B colors n C ths s the content of quaton 2. Now take a color c = {v, w } C and suppose that c has multplcty µ c. Then there exst vectors x S for = 1, 2,..., µ c such that c s the color between x, v and x, w. Consder the colors of the two pars x 1, v, x 2, v and x 1, v, x 2, w n C. These are cp x 1, x 2, η p 1 v, v = c 1,2, 0 C and cp x 1, x 2, η p 1 v, w = c 1,2, η p 1 c C, respectvely, where c 1,2 C B here we abuse notaton and defne η p 1 c = η p 1 v, w, whch s allowed snce the rght-hand-sde s symmetrc n the two nput coordnates. Note that by the nductve hypothess, there are at least µ c 1 dstnct colors of the form c,j for dstnct pars of ndces and j. Hence, by consderng these colors, we add colors of the types c,j, 0 and c,j, η p 1 c for at least µ c 1 dstnct colors c,j C B. ven f one of these two colors equals the color c,j,? counted above, we have added at least µ c 1 colors to C by consderng the color c C. Now consder another color c 1 C. Ths color adds a further µ c 1 colors to C as long as 1 η p 1 c η p 1 c 1. Therefore, f we can somehow guarantee that η p 1c s dstnct for all c, then we have C C B + µ c 1, c C whch proves 4, snce C B S 1 by the nductve hypothess. Hence, t would be helpful to have dstnct η p 1 c for each c C. ven though we cannot always guarantee ths, we can show that there exsts a resoluton n whch the correspondng fact does hold. Ths wll be explaned n more detal n Secton 5. 4 Propertes of the colorng n ths secton, we collect some useful facts about the colorng functons c d. Before lstng these propertes, we ntroduce the formal framework that we wll use to descrbe them. 4.1 Refnement of functons For a functon f : A B, let Π f = {f 1 b : b fa}. Thus, Π f s a partton of A nto sets whose elements map by f to the same element n B. For two functons f and g defned over the 10

11 same doman, we say that f refnes g f Π f s a refnement of Π g. Ths defnton s equvalent to sayng that fa = fa mples that ga = ga and s also equvalent to sayng that there exsts a functon h for whch g = h f. The term f refnes g s also referred to as g factors through f n category theory. Ths formalzes the concept that f contans more nformaton than g. For two functons f and g defned over the same doman A, let f g be the functon defned over A where f ga = fa, ga. The followng proposton collects several basc propertes of refnements of functons whch wll be useful n the proof of the man theorem. Proposton 4.1. Let f 1, f 2, f 3 and f 4 be functons defned over the doman A. dentty f 1 refnes f 1. Transtvty f f 1 refnes f 2 and f 2 refnes f 3, then f 1 refnes f 3. f f 1 refnes f 3, then f 1 f 2 refnes f 3. v f f 1 refnes both f 2 and f 3, then f 1 refnes f 2 f 3. v f f 1 refnes f 3 and f 2 refnes f 4, then f 1 f 2 refnes f 3 f 4. v f f 1 refnes f 2, then, for all A A, we have f 1 A f 2 A. Proof. Let Π = Π f for = 1, 2, 3. Ths s trval snce Π 1 refnes Π 1. f f 1 refnes f 2 and f 2 refnes f 3, then Π 1 refnes Π 2 and Π 2 refnes Π 3. Therefore, Π 1 refnes Π 3 and f 1 refnes f 3. Snce f 1 f 2 clearly refnes f 1, ths follows from. v f f 1 a = f 1 a, then f 2 a = f 2 a and f 3 a = f 3 a. Hence, f 2 f 3 a = f 2 f 3 a and we conclude that f 1 refnes f 2 f 3. v By, f 1 f 2 refnes both f 3 and f 4. Therefore, by v, f 1 f 2 refnes f 3 f 4. v For = 1, 2, let Π A = {X A : X Π, X A } and note that f A = Π A. Snce Π 1 s a refnement of Π 2, we see that Π 1 A s a refnement of Π 2 A. Therefore, t follows that f 1 A = Π 1 A Π 2 A = f 2 A, as requred. Refnements arse n our proof because we often consder colorngs wth less nformaton than the full colorng. n the outlne above, we consdered several dfferent sets of colors, namely, Λ, Λ, C and C and we clamed wthout proof that C Λ and C Λ. f we can show that Λ s a refnement of C and Λ s a refnement of C, then these nequaltes follow from Proposton 4.1 v above. 11

12 4.2 Propertes of the colorng We developed our formal framework for a rgorous treatment of the followng two lemmas. t may be helpful at ths stage to recall the defntons of η d, ξ d and c d from Subsecton 2.3. Lemma 4.2. Suppose that α, α and d are ntegers wth d 0 and 1 α α. Then the followng hold where all functons are consdered as defned over {0, 1} α {0, 1} α : η d refnes η d π α π α. ξ d refnes ξ d π α π α. c d refnes c d π α π α. Proof. The case α = α s trval so we assume that α < α. Let v and w be vectors n {0, 1} α and let v = π α v and w = π α w. We wll show that one can compute the value of η d v, w based only on the value of η d v, w. Ths clearly mples the desred concluson. f η d v, w = 0, then v = w and t follows that η d v, w = 0. Assume then that η d v, w =, {v d, w d } for some ndex and blocks v d, w d of resoluton d. Let j be the frst coordnate n whch the two vectors v d and w d dffer. Then the frst coordnate x note that 1 x α n whch v and w dffer s x = 1 r d + j and satsfes 1 r d < x mn{ r d, α}. Note that the values of and j can be deduced from η d v, w and hence x can as well. t thus suffces to verfy that η d v, w can be computed usng only α, α, r d, x,, v d and w d. f α > r d, then we have η d v, w = η d v, w =, {v d, w d } and the clam s true. On the other hand, f α r d, then there are two cases. f α < x, then we have v = w. Therefore, η d v, w = 0 and the clam holds for ths case as well. The fnal case s when x α r d. n ths case, we see that and the clam holds. η d v, w = {, π [α 1r d ]v d Let v and w be two vectors n {0, 1} α. Then }, π [α 1r d ]w d ξ d v, w = η d v d+1 1, w d+1 1, η d v d+1 2, w d+1 2,..., η d v d+1 a+1, wd+1 a+1, for some nteger a 0. Let v = π α v and w = π α w. Suppose that j 1r d+1 < α jr d+1. Then note that the j-th block of resoluton d + 1 of v s π [α j 1r d+1 ]v d+1 j and that of w s π [α j 1r d+1 ]w d+1 j. Then ξ d v, w conssts of j coordnates, where for 1 < j the -th coordnate s dentcal to the -th coordnate of ξ d v, w and, for = j, the j-th coordnate s η d π [α j 1r d+1 ] π [α j 1r d+1 ]v d+1 j, w d+1 j. 12

13 Thus the functon ξ d refnes ξ d π α π α coordnate by coordnate by part of ths lemma. Hence, by Proposton 4.1v, we see that ξ d refnes ξ d π α π α. Ths follows from c d = ξ d ξ 0, part of ths lemma and Proposton 4.1v. Lemma 4.2 seems ntutvely obvous and mght even seem trval at frst sght, but a moment s thought reveals the fact that t s nontrval. To see ths, consder the functon h d v, w = {v d, w d }, whch s the projecton to the second coordnate of η d v, w. Then the functon h d fals to satsfy Lemma 4.2. Moreover, f the functons ξ d and c d were bult usng h d nstead of η d, these would also fal to satsfy the clam of Lemma 4.2. The next lemma completes the proof of one of the promsed clams, namely, that Λ or, rather, a generalzaton thereof refnes C. Lemma 4.3. Suppose that postve ntegers d, p, α and α are gven such that 1 d p + 1 and α s the maxmum nteger less than α dvsble by r d. Let γ d be the functon whch takes a par of vectors v, w {0, 1} α as nput and outputs γ d v, w = c p v, w, η d 1 v, w, where v = v, v and w = w, w for v, w {0, 1} α and v, w {0, 1} α α. Then c p {0,1} α {0,1} α refnes γ d. Proof. For brevty, we restrct the functons to the set {0, 1} α {0, 1} α throughout the proof. By Lemma 4.2, we know that c p refnes c p π α π α and hence c p refnes the frst coordnate of γ d. On the other hand, snce α s the maxmum nteger less than α dvsble by r d, the term η d 1 v, w forms the last coordnate of the vector ξ d 1 v, w. Hence, by Proposton 4.1, ξ d 1 refnes η d 1 v, w. By the defnton of c p and Proposton 4.1, we know that c p refnes ξ d 1. Therefore, by transtvty Proposton 4.1, we see that c p refnes η d 1 v, w. Thus, c p refnes both coordnates of γ d and hence, by Proposton 4.1v, we see that c p refnes γ d. 5 Proof of the man theorem n ths secton we prove Theorem 2.2, whch asserts that for all α 1 and p 1, the edge-colorng of the complete graph on the vertex set {0, 1} α gven by c p s a p + 3, p + 2-colorng. We wll prove by nducton on α that every set S wth S p + 3 receves at least S 1 dstnct colors. The base case s when α r p. n ths case, for two dstnct vectors v, w {0, 1} α, we have ξ p v, w = η p v, w = 1, {v, w}. Hence, for a gven set S {0, 1} α, the edges wthn ths set are all colored wth dstnct colors, thereby mplyng that at least S 2 S 1 colors are used. 13

14 Now suppose that α > r p s gven and the clam has been proved for all smaller values of α. Let S {0, 1} α be a gven set wth S p + 3. For each 1 d p, let α d be the largest nteger less than α whch s dvsble by r d. Note that snce r d 1 dvdes r d for all 1 d p, we have For 1 d p, defne sets Λ d and and Λ d α p α p 1 α 1. as Λ d = { c p v, w : π αd v π αd w, v, w S } Λ d = { c p v, w : π αd v = π αd w, v w, v, w S }. Snce α d s dvsble by r d, f π αd v = π αd w, then the frst α d r d coordnates of ξ d 1 v, w wll all be zero. On the other hand, f π αd v π αd w, then ths s not the case. Snce ξ d 1 s part of c p, ths mples that Λ d + Λ d Λ d Λ d. t therefore suffces to prove that Λd =. Hence, for all d, the number of colors wthn S s exactly + Λ d S 1 for some ndex d. We would lke to extract only the mportant nformaton from the colors n Λ d and Λ d. For d for each 1 d p and a gven par of vectors v, w S, let v = v d, v d and w = w d, w v d, w d {0, 1}α d and v d, w d {0, 1}α α d. Defne the sets C d and C d C d = { cp v d, w d, η d 1v d, w d : v d w d, v, w S } as and C d = { {v d, w d } : v d = w d, v d w d, v, w S }. By Lemma 4.3 and Proposton 4.1v, we see that C d Λ d. We also have C d Λd. To see ths, suppose that a color {v d, w d } Cd comes from a par of vectors v = v d, v d and w = w d, w d n S. Snce v d = w d and α d s dvsble by r d, the functon η d appled to the last par of blocks of resoluton d + 1 of v and w s equal to, {v } for some nteger. Therefore, the last coordnate of ξ d v, w has value, {v suffces to prove that C d + C d d, w d d, w d }. Ths mples that Cd S 1 for some ndex 1 d p. Λd. Hence, t now Assume for the sake of contradcton that we have C d + C d S 2 for all 1 d p. The followng s the key ngredent n our proof. Clam 5.1. f C p + C p S 2, then there exsts an ndex d such that η d 1c s dstnct for each c C d. The proof of ths clam wll be gven later. Let d be the ndex guaranteed by ths clam and let C = C d, C = C d. Abusng notaton, for two vectors v, w S, we wll from now on refer to the color between v and w as the correspondng color n C or C. 14

15 Let S = π αd S and, for a vector v S, let T v = {v : π αd v = v, v S}. Note that the sets T v form a partton of S. Therefore, T v = S. 5 v S Let C B be the set of colors whch appear wthn the set S under the colorng c p. Snce S {0, 1} α d and α d < α, the nductve hypothess mples that C B S 1. 6 For a color c C, let µ c be the number of unordered pars of vectors v, w such that c s the color between v and w. Note that Tv µ c = T v c C v S v S Together wth the three equatons above, the followng bound on C, whose proof we defer for a moment, yelds a contradcton. C C B + c C µ c 1. 8 ndeed, f ths nequalty holds, then, by 8, 6 and 7, respectvely, we have C + C S 1 + µ c 1 + C = S 1 + c C S 1 + v S T v 1 = v S T v 1. c C µ c By 5, we see that the rght hand sde s equal to S 1. Therefore, we obtan C + C S 1, whch contradcts the assumpton that C + C S 2. To prove 8, we examne the nteracton between the three sets of colors C, C B and C. Note that each color c C s of the form c = c,? for some c C B, where the queston mark? stands for an unspecfed coordnate. Ths fact already gves the trval bound C C B. To obtan 8, we mprove ths nequalty by consderng the? part of the color and ts relaton to colors n C. Take a color c = {v, w } C and suppose that c has multplcty µ c 2. Then there exst vectors x, y S such that x, v, x, w T x and y, v, y, w T y. Consder the color of the pars x, v, y, v and x, v, y, w n C. These colors are of the form cp x, y, η d 1 v, v = c p x, y, 0 C and cp x, y, η d 1 v, w = c p x, y, η d 1 c C. 15

16 Here we abuse notaton and defne η d 1 c = η d 1 v, w, whch s allowed snce the rght-handsde s symmetrc n the two nput coordnates. Therefore, havng a color c wth µ c 2 already mples that C C B + 1. We carefully analyze the gan comng from these pars for each color n C. To ths end, for each x S, we defne { C,x = {v, w } : x, v, x, w T x, v w }. For each c C B, we wll count the number of colors of the form c,? C. There are two cases. Case 1 : For all x, y S wth c p x, y = c, C,x C,y =. Apply the trval bound assertng that there s at least one color of the form c,? n C. Case 2 : There exsts a par x, y S wth c p x, y = c such that C,x C,y. f we have c C,x C,y for some x, y S wth c p x, y = c, then, by the observaton above, we have both c, 0 and c, η d 1 c n C. Ths shows that the number of colors n C of the form c,? s at least {c, 0} { c, η d 1 c : x, y S, c p x, y = c, c C,x C,y }. By Clam 5.1, the functon η d 1 s njectve on C and thus the above number s equal to 1 + { c : x, y S, c p x, y = c, c C,x C,y }. By combnng cases 1 and 2, we see that the number of colors n C satsfes C C B + { c : x, y S, c p x, y = c, c } C,x C,y c C B = C B + c C { c : x, y S, c p x, y = c, c C,x C,y }. For a fxed color c C, there are precsely µ c vectors x S for whch the color c s n C,x. Hence, by the nducton hypothess, for each fxed c, we have { c : x, y S, c p x, y = c, c C,x C,y } µc 1. Thus we obtan whch s 8. C C B + c C µ c 1, 5.1 Proof of Clam 5.1 Clam 5.1 asserts that there exsts an ndex d such that η d 1 c s dstnct for each c C d. 16

17 t wll be useful to consder the functon h d, whch s defned as follows: for dstnct vectors v and w, defne h d v, w = {v d, w d }, where v d, w d are the frst par of blocks of resoluton d for whch v d w d. Also, defne h d v, v = 0 for all vectors v. Note that we can also defne h d over unordered pars {v, w} of vectors as h d {v, w} = h d v, w, snce h d v, w = h d w, v for all pars v and w. Throughout the subsecton, by abusng notaton, we wll be applyng h d to both ordered and unordered pars wthout further explanaton. Recall that η d v, w =, {v d, w d } and η d v, v = 0 and, therefore, η d refnes h d both consdered as functons over the doman C d. Hence, to prove the clam, t suffces to prove that h d 1c s dstnct for each c C d. Another mportant observaton s that for all 1 d p, we can redefne the sets C d as C d = { h d v, w : π αd v = π αd w, v w, v, w S }. We frst prove that there s a certan monotoncty between the sets C d for 1 d p. Clam 5.2. For all d satsfyng 2 d p, there exsts an njectve map j d : C d 1 maps {x, y} C d 1 to { } j d x, y = v, x, v, y C d, C d whch for some vector v {0, 1} α d 1 α d dependng on the color {x, y}. Furthermore, h d 1 j d s the dentty map on C d 1. Proof. Take a color {x, y} C d 1 and assume that {x, y} = h d 1 v x, v y for v x, v y S. By the defnton of C d 1, we may take v x and v y of the form v x = v 0, x and v y = v 0, y, for some vector v 0 {0, 1} α d 1. Fx an arbtrary such par v x, v y for each {x, y} C d 1. Let v 0 = v 1, v 2 for v 1 {0, 1} α d and v 2 {0, 1} α d 1 α d. Then v x = v 1, v 2, x and v y = v 1, v 2, y. Snce π αd v x = v 1 = π αd v y, we see that h d v x, v y = { } v 2, x, v 2, y C d. Defne j d x, y = h d v x, v y and note that the range of j d s ndeed C d. Moreover, snce v 2 s a vector of length α d 1 α d whch s dvsble by r d 1, we see that h d 1 j d x, y = h d 1 v 2, x, v 2, y = {x, y}. The clam follows. 17

18 n partcular, Clam 5.2 mples that C 1 C2 Cp. f C 1 1, then d = 1 trvally satsfes the requred condton. Hence, we may assume that C 1 2. On the other hand, recall that we are assumng that Cp + C p S 2 p + 1. f = 0, then there exsts at most one element v p π αp S and all elements of S are of the form C p v p, x for some x {0, 1} α αp. But then C p S S 1, 9 2 contradctng our assumpton. Therefore, we may assume that C p 1, from whch t follows that C p p. Hence, 2 C 1 C2 Cp p. f p = 1, ths s mpossble. f p 2, then, by the pgeonhole prncple, there exsts an ndex d such that C d 1 = C d. For ths ndex, the map j d defned n Clam 5.2 becomes a bjecton. Then, snce h d 1 j d s the dentty map on C d 1 proves the clam., we see that h d 1 c are dstnct for all c C d. Ths 6 Concludng Remarks 6.1 Better than p + 3, p + 2-colorng p+4 Let r = 2. We can n fact prove that c p s a p + r + 1, p + r -colorng. Ths mprovement comes from explotng the slackness of the nequalty 9 used n Subsecton 5.1. To see ths, we replace the bound on S by S p + r + 1 n the proof gven above. Snce we have already proved the result for S p + 3, we may assume that S p + 4. f C p r 1, then we have C p S 2 Cp p and we can proceed as n the proof above. We may therefore assume that C p < r 1. Let S p = π αp S. Then, snce S p 1 C p < r 1, we know that S p < r. Snce v S p π 1 α p v = S, there exsts a v S p such that π 1 α p v S S p. Note that every par of vectors w 1, w 2 π 1 α p v gves a dstnct emergng color. Moreover, by the nductve hypothess, we have at least S p 1 18

19 nherted colors. Hence, the total number of colors n the colorng c p wthn the set S s at least whch, snce π 1 α S p 1 + p v 2 S p S 2 S p p + 4 S p < r = 2 S 2, S S p 1, s mnmzed when S p s maxmzed. Thus the number of colors wthn the set S s at least Ths concludes the proof. S S S 2 = S Usng fewer colors Recall that the colorng c p was bult from the functons where s the mnmum ndex for whch v d the functon η d v, w =, {v d, w d }, h d v, w = { v d w d. The functon η d can n fact be replaced by, w d } note that ths s the functon used n Secton 5.1. n other words, even f we replace all occurrences of η d wth h d n the defnton of c p, we can stll show that c p s a p + 3, p + 2-colorng. Moreover, there exsts a constant a p such that the colorng of the complete graph on n vertces defned n ths way uses only aplog n1 1/p+1 2 colors. That s, we gan a log log n factor n the exponent compared to Theorem 2.1. The tradeoff s that the proof s now more complcated, the chef dffculty beng to fnd an approprate analogue of Lemma 4.2 whch works when η d s replaced by h d. 6.3 Top-down approach There s another way to understand our colorng as a generalzaton of Mubay s colorng. Recall that Mubay s colorng s gven as follows: for two vectors v, w [m] t satsfyng v = v 1,..., v t and w = w 1,..., w t, let cv, w = {v, w }, a 1, a 2,..., a t, where s the mnmum ndex for whch v w and a j = 0 f v j = w j and a j = 1 f v j w j. 19

20 Suppose that we are gven postve ntegers t 1 and t 2. For two vectors v, w [m] t 1t 2, let v = v 1 1,..., v1 t 2 and w = w 1 1,..., w1 t 2 for vectors v 1 [m] t 1 and w 1 [m] t 1. Defne the colorng c 2 as c 2 v, w = {v 1, w 1 }, cv 1 1, w1 1,..., cv1 t 2, w 1 t 2, where s the mnmum ndex for whch v 1 w 1. Note that ths can also be understood as a varant of c, where we record more nformaton n the a 1,..., a t part of the vector ths s a top-down approach and the prevous defnton s a bottom-up approach. The colorng c 2 s essentally equvalent to c 2 defned n Secton 6.2 above and can be further generalzed to gve a colorng correspondng to c p for p 3. However, the proof agan becomes more techncal for ths choce of defnton. One advantage of defnng the colorng usng ths top-down approach s that t becomes easer to see why the colorng c p on K n2 contans the colorng c p on K n1, where n 1 < n 2, as an nduced colorng. To see ths n the example above, suppose that n 1 = m t 1t 2 and n 2 = n s 1s 2 for m n, t 1 s 1 and t 2 s 2. Then the natural njecton from [m] to [n] extends to an njecton from [m] t 1 to [n] s 1 and then to an njecton from [m] t 1t 2 to [n] s 1s 2. Ths njecton shows that the colorng c 2 on K n2 contans the colorng c 2 on K n1 as an nduced colorng. As n Secton 2.1, t then follows that c 2 and thus c 2 fals to be a q, q ε -colorng for large enough q. Smlarly, for all fxed p 3, we can show that c p fals to be a q, q ε -colorng for large enough q. 6.4 Stronger propertes We can show see [1] that Mubay s colorng, dscussed n Secton 2.1, actually has the followng stronger property: for every par of colors, the graph whose edge set s the unon of these two color classes has chromatc number at most three prevously, we only establshed the fact that the clque number s at most three. We suspect that ths property can be generalzed. Queston 6.1. Let p 4 be an nteger. Does there exst an edge-colorng of the complete graph K n wth n o1 colors such that the unon of every p 1 color classes has chromatc number at most p? We do not know whether our colorng has ths property or not. 6.5 Lower bound Some work has also been done on the lower bound for fn, p, p 1. As mentoned n the ntroducton, for p = 3 t s known that c log n log log n fn, 3, 2 c log n. For p = 4, the gap between the lower and upper bounds s much wder. The well-known bound r k 4 k ck on the multcolor Ramsey number of K 4 translates to fn, 4, 3 c log n log log n, whle Mubay s colorng gves an upper bound of fn, 4, 3 e c log n. The lower bound has been mproved, frst by Kostochka and Mubay 20

21 log n log log log n [7], to fn, 4, 3 c current best known bound. and then, by Fox and Sudakov [6], to fn, 4, 3 c log n, whch s the For p 5, we can obtan a smlar lower bound from the followng formula, vald for all p and q. f nfn, p 1, q 1, p, q fn, p 1, q To prove ths formula, put N = nfn, p 1, q 1 and consder an edge-colorng of K N wth fewer than fn, p 1, q 1 colors. t suffces to show that there exsts a set of p vertces whch uses at most q 1 colors on ts edges. f fn, p 1, q 1 = 1, then the nequalty above s trvally true. f not, then for a fxed vertex v, there exsts a set V of at least N 1 fn,p 1,q 1 1 n vertces adjacent to v by the same color. Snce the edges wthn the set V are colored by fewer than fn, p 1, q 1 colors, the defnton of fn, p 1, q 1 mples that we can fnd a set X of p 1 vertces wth at most q 2 colors used on ts edges. t follows that the set X {v} s a set of p vertces wth at most q 1 colors used on ts edges. The clam follows. From 10 and the lower bound fn, 4, 3 c log n, one can deduce that fn, p, p o1fn, 4, 3 c + o1 log n for all p 5. On the other hand, snce the best known upper bound on fn, p, p 1 s fn, p, p plog n1 1/p 2 log log n, the gap between the upper and lower bounds gets wder as p gets larger. t would be nterestng to know whether ether bound can be substantally mproved. n partcular, the followng queston seems mportant. Queston 6.2. For p 5, can we gve better lower bounds on fn, p, p 1 than the one whch follows from fn, 4, 3? References [1] D. Conlon, J. Fox, C. Lee and B. Sudakov, On the grd Ramsey problem and related questons, n preparaton. [2] D. chhorn and D. Mubay, dge-colorng clques wth many colors on subclques, Combnatorca , [3] P. rdős, Problems and results on fnte and nfnte graphs, n Recent advances n graph theory Proc. Second Czechoslovak Sympos., Prague, 1974, , Academa, Prague, [4] P. rdős, Solved and unsolved problems n combnatorcs and combnatoral number theory, n Proceedngs of the Twelfth Southeastern Conference on Combnatorcs, Graph Theory and Computng, Vol. Baton Rouge, La., 1981, Congr. Numer ,

22 [5] P. rdős and A. Gyárfás, A varant of the classcal Ramsey problem, Combnatorca , [6] J. Fox and B. Sudakov, Ramsey-type problem for an almost monochromatc K 4, SAM J. Dscrete Math /09, [7] A. Kostochka and D. Mubay, When s an almost monochromatc K 4 guaranteed? Combn. Probab. Comput , [8] D. Mubay, dge-colorng clques wth three colors on all 4-clques, Combnatorca , [9] J. Nešetřl and M. Rosenfeld,. Schur, C.. Shannon and Ramsey numbers, a short story, Dscrete Math , [10] V. Rosta, Ramsey theory applcatons, lectron. J. Combn , Research Paper 89, 48 pp. [11] G. N. Sárközy and S. M. Selkow, On edge colorngs wth at least q colors n every subset of p vertces, lectron. J. Combn , Research Paper 9, 6pp. [12] G. N. Sárközy and S. M. Selkow, An applcaton of the regularty lemma n generalzed Ramsey theory, J. Graph Theory , [13]. Schur, Über de Kongruenz xm + y m = z m mod p, Jber. Deutsch. Math.-Veren ,