The Edős-Hajnal conjectue fo ainbow tiangles Jacob Fox Andey Ginshpun János Pach Abstact We pove that evey 3-coloing of the edges of the complete gaph on n vetices without a ainbow tiangle contains a set of ode Ω n 1/3 log n which uses at most two colos, and this bound is tight up to a constant facto This veifies a conjectue of Hajnal which is a case of the multicolo genealization of the well-known Edős-Hajnal conjectue We futhe establish a genealization of this esult Fo fixed positive integes s and with s, we detemine a constant c,s such that the following holds Evey -coloing of the edges of the complete gaph on n vetices without a ainbow tiangle contains a set of ode Ω n 1/ss 1 log n c,s which uses at most s colos, and this bound is tight apat fom the implied constant facto The poof of the lowe bound utilizes Gallai s classification of ainbow-tiangle fee edge-coloings of the complete gaph, a new weighted extension of Ramsey s theoem, and a discepancy inequality in edge-weighted gaphs The poof of the uppe bound uses Edős lowe bound on Ramsey numbes by consideing lexicogaphic poducts of -edge-coloings of complete gaphs without lage monochomatic cliques 1 Intoduction A classical esult of Edős and Szekees [8], which is a quantitative vesion of Ramsey s theoem [17], implies that evey gaph on n vetices contains a clique o an independent set of ode at least 1 log n In the othe diection, Edős [6] showed that a andom gaph on n vetices almost suely contains no clique o independent set of ode log n An induced subgaph of a gaph is a subset of its vetices togethe with all edges with both endpoints in this subset Thee ae seveal esults and conjectues indicating that gaphs which do not contain a fixed induced subgaph ae highly stuctued In paticula, the most famous conjectue of this sot by Edős and Hajnal [7] says that fo each fixed gaph H thee is ɛ = ɛh > 0 such that evey gaph G on n vetices which does not contain a fixed induced subgaph H has a clique o independent set of ode n ɛ This is in stak contast to geneal gaphs, whee the ode of the lagest guaanteed clique o independent set is only logaithmic in the numbe of vetices Thee ae now seveal patial esults on the Edős-Hajnal conjectue Edős and Hajnal [7] poved that fo each fixed gaph H thee is ɛ = ɛh > 0 such that evey gaph G on n vetices which does Depatment of Mathematics, Massachusetts Institute of Technology, Cambidge, MA 0139-4307 Email: fox@mathmitedu Reseach suppoted by a Simons Fellowship, by NSF gant DMS-1069197, by a Sloan Foundation Fellowship, and by an MIT NEC Copoation Awad Depatment of Mathematics, Massachusetts Institute of Technology, Cambidge, MA 0139-4307 Email: aginshp@mathmitedu Reseach suppoted by a National Physical Science Consotium Fellowship EPFL, Lausanne and Couant Institute, New Yok, NY Suppoted by Hungaian Science Foundation EuoGIGA Gant OTKA NN 1009, by Swiss National Science Foundation Gants 0000-144531 and 0001-137574, and by NSF Gant CCF- 08-307 Email: pach@cimsnyuedu 1
not contain an induced copy of H has a clique o independent set of ode e ɛ log n Fox and Sudakov [9], stengthening an ealie esult of Edős and Hajnal, poved that fo each fixed gaph H thee is ɛ = ɛh > 0 such that evey gaph G on n vetices which does not contain an induced copy of H has a balanced complete bipatite gaph o an independent set of ode n ɛ All gaphs on at most fou vetices ae known to satisfy the Edős-Hajnal conjectue, and Chudnovsky and Safa [4] poved it fo the 5-vetex gaph known as the bull Alon, Pach, and Solymosi [1] poved that if H 1 and H satisfy the Edős-Hajnal conjectue, then fo evey v of H 1, the gaph fomed fom H by substituting v by a copy of H satisfies the Edős-Hajnal conjectue The ecent suvey [3] discusses many futhe elated esults on the Edős-Hajnal conjectue A natual estatement of the Edős-Hajnal conjectue is that fo evey fixed ed-blue edge-coloing χ of a complete gaph, thee is an ɛ = ɛχ > 0 such that evey ed-blue edge-coloing of the complete gaph on n vetices without a copy of χ contains a monochomatic clique of ode n ɛ Indeed, fo the gaphs H and G, we can colo the edges ed and the nonadjacent pais blue Edős and Hajnal also poposed studying a multicolo genealization of thei conjectue It states that fo evey fixed k-coloing of the edges of χ of a complete gaph, thee is an ɛ = ɛχ > 0 such that evey k-coloing of the edges of the complete gaph on n vetices without a copy of χ contains a clique of ode n ɛ which only uses k 1 colos They poved a weake estimate, eplacing n ɛ by e ɛ log n Note that the case of two colos is what is typically efeed to as the Edős-Hajnal conjectue Hajnal [14] conjectued the following special case of the multicolo genealization of the Edős-Hajnal conjectue holds Thee is ɛ > 0 such that evey 3-coloing of the edges of the complete gaph on n vetices without a ainbow tiangle that is, a tiangle with all its edges diffeent colos, contains a set of ode n ɛ which uses at most two colos We pove Hajnal s conjectue, and futhe detemine a tight bound on the ode of the lagest guaanteed -coloed set in any such coloing A Gallai coloing is a coloing of the edges of a complete gaph without ainbow tiangles, and a Gallai -coloing is a Gallai coloing that uses colos Theoem 11 Evey Gallai 3-coloing on n vetices contains a set of ode Ωn 1/3 log n which uses at most two colos, and this bound is tight up to a constant facto To give an uppe bound, we use lexicogaphic poducts We will let [m] = {1,, m} denote the set consisting of the fist m positive integes Definition 1 Given edge-coloings F 1 of K m1 and F of K m using colos fom R, the lexicogaphic poduct coloing F 1 F of EK m1 m is defined on any edge e = {u 1, v 1, u, v } whee we take the vetex set of K m1 m to be [m 1 ] [m ] to be F 1 u 1, u if u 1 u, and othewise v 1 v and it is defined to be F v 1, v That is, thee ae m 1 disjoint copies of F and they ae connected by edge colos defined by F 1 The uppe bound in Theoem 11 is obtained by taking the lexicogaphic poduct of thee -edgecoloings of the complete gaph on n 1/3 vetices, whee each pai of colos is used in one of the coloings, and the lagest monochomatic clique in each of the coloings is of ode Olog n A simple lemma in the next section shows that, in a lexicogaphic poduct coloing F = F 1 F, the lagest set of vetices using only colos ed and blue fo example in F has size equal to the poduct of the size of
the lagest set of vetices using only colos ed and blue in F 1 with the size of the lagest set of vetices using only colos ed and blue in F Fo any set S of two of the thee colos, the lagest such set has ode On 1/3 Olog nolog n = On 1/3 log n In the othe diection, we will utilize the following impotant stuctual esult of Gallai [11] on edge-coloings of complete gaphs without ainbow tiangles Lemma 13 An edge-coloing F of a complete gaph on a vetex set V with V is a Gallai coloing if and only if V may be patitioned into nonempty sets V 1,, V t with t > 1 so that each V i has no ainbow tiangles unde F, at most two colos ae used on the edges not intenal to any V i, and the edges between any fixed pai V i, V j use only one colo Futhemoe, any such substitution of Gallai coloings fo vetices of a -edge-coloing of a complete gaph K t yields a Gallai coloing Gallai coloings natually aise in seveal aeas including in infomation theoy [15], in the study of patially odeed sets, as in Gallai s oiginal pape [11], and in the study of pefect gaphs [] Thee ae now a vaiety of papes which conside Ramsey-type poblems in Gallai coloings see, eg, [5, 10, 1, 13] Howeve, these woks mainly focus on finding vaious monochomatic subgaphs in such coloings Because it may be of independent inteest to the eade, we fist pesent a paticulaly simple appoach that will pove Hajnal s conjectue, but will not give tight bounds A gaph is a cogaph if it has at most one vetex, o if it o its complement is not connected, and all of its induced subgaphs have this popety In othe wods, the family of cogaphs consists of all those gaphs that can be obtained fom an isolated vetex by successively taking the disjoint union of two peviously constucted cogaphs, G 1 and G, o by the join of them that we get by adding all edges between G 1 and G It was shown by Seinsche [18] that cogaphs ae pecisely those gaphs which do not contain the path with thee edges as an induced subgaph It is easy to check by induction that evey cogaph is a pefect gaph, that is, the chomatic numbe of evey induced subgaph is equal to its clique numbe Poposition 14 In any Gallai 3-coloing of a complete gaph, thee is an edge-patition of the complete gaph into thee -coloed subgaphs, each of which is a cogaph Poof: This follows fom Gallai s stuctue theoem by induction on the numbe of vetices The esult is tivial fo edge-coloings of complete gaphs with fewe than two vetices, which seves as the base case Using Lemma 13, we get a nontivial vetex patition of the Gallai 3-coloing of the complete gaph into pats V 1,, V t such that only two colos appea between the pats By the induction hypothesis, we can patition the edge-set of the complete gaph on V i into thee cogaphs, each which is two-coloed Fo the two colos that go between the pats, we take the gaph which is the join of the cogaphs in each V i, that is, add all edges between the pats, and fo each of the othe two pais of colos, we just take the disjoint union of the cogaphs of those two colos fom each pat Since the join o disjoint union of cogaphs ae cogaphs, this completes the poof by induction The following coollay veifies Hajnal s conjectue and, apat fom the two logaithmic factos, gives the lowe bound in Theoem 11 Coollay 15 Evey Gallai 3-coloing of EK n contains a -coloed clique with at least n 1/3 vetices 3
Poof: Indeed, applying Poposition 14, eithe the fist cogaph which is -coloed contains a clique of ode n 1/3 in which case we ae done, o it contains an independent set of ode n /3 In the latte case, this independent set of ode n /3 in the fist cogaph contains in the second cogaph a clique of ode n 1/3 o an independent set which is a clique in the thid cogaph of ode n 1/3 We thus get a clique of ode n 1/3 in one of the thee cogaphs, which is a -coloed set Impoving the lowe bound futhe to Theoem 11 appeas to be consideably hade, and uses a diffeent poof technique, elying on a weighted vesion of Ramsey s theoem and a caefully chosen induction agument The weighted vesion of Ramsey s theoem shows that if each vetex of a complete gaph on t vetices is given a positive ed weight and a positive blue weight whose poduct is one, then in any ed-blue edge-coloing of K t, thee is a ed clique S and a blue clique U such that the poduct of the ed weight of S the sum of the ed weights of the vetices in S and the blue weight of U the sum of the blue weights of the vetices in U is Ω log t Note that this extends the quantitative vesion of Ramsey s theoem as the case in which all the ed and blue weights ae one implies that thee is a monochomatic clique of ode Ωlog t We futhe conside a natual genealization of this poblem to moe colos, and give a tight bound in the next theoem In ode to state the esult moe succinctly, we intoduce some notation: fo positive integes and s with s, let 1 if 1 = s < o if s = 1 and is even; s s if 1 < s < 1; c,s = 1 + 3 if s = 1 and is odd; 0 if s = Theoem 16 Let and s be fixed positive integes with s Evey -coloing of the edges of the complete gaph on n vetices without a ainbow tiangle contains a set of ode Ωn s / log c,s n which uses at most s colos, and this bound is tight apat fom the constant facto We next give a bief discussion of the poof of Theoem 16 The case s = is tivial as the complete gaph uses at most colos The case s = 1 is easy Indeed, in this case, by the Edős-Szekees bound on Ramsey numbes fo colos, thee is a monochomatic set of ode Ωlog n, whee the implied positive constant facto depends on In the othe diection, we give a constuction which we conjectue is tight The Ramsey numbe t is the minimum n such that evey -coloing of the edges of the complete gaph on n vetices contains a monochomatic clique of ode t The bounds mentioned in the beginning of the intoduction give t/ t t fo t Fo even, conside a lexicogaphic poduct of / coloings, each a -edge coloing of the complete gaph on t 1 vetices with no monochomatic K t This gives a Gallai -coloing of the edges of the complete gaph on t 1 / vetices with no monochomatic clique of ode t A simila constuction fo odd gives a Gallai -coloing of the edges of the complete gaph on t 1 t 1 1/ vetices with no monochomatic clique of ode t The following conjectue which states that these bounds ae best possible seems quite plausible It was veified by Chung and Gaham [5] in the case t = 3 Conjectue 17 Let N, t = t 1 / fo even and N, t = t 1 t 1 1/ fo odd 4
Fo n > N, t, evey -coloing of the edges of the complete gaph on n vetices has a ainbow tiangle o a monochomatic K t Having veified the easy cases s = 1 and s = of Theoem 16, fo the est of the pape, we assume 1 < s < A natual uppe bound on the size of the lagest set using at most s colos comes fom the following constuction We will let [] be the set of colos Conside the complete gaph on [], whee each edge P gets a positive intege weight n P such that the poduct of all n P is n Fo each edge P of this complete gaph, we conside a -coloing c P of the edges of the complete gaph on n P vetices using the colos in P and whose lagest monochomatic clique has ode Olog n P, which exists by Edős lowe bound [6] on Ramsey numbes We then conside the Gallai -coloing c of the complete gaph on n vetices which is the lexicogaphic poduct of the coloings of the fom cp Fo each set S of colos, the lagest set of vetices in this edge-coloing of K n using only colos in S has ode Olog n P P S n P P S =1 The ode of the lagest set using at most s colos in coloing c is thus the maximum of the above expession ove all subsets S of colos of size s Theefoe, we want to choose the vaious n P to minimize this maximum Fo s < 1, we give a second moment agument which shows that the best choice is essentially that the n P ae all equal, ie, n P = n 1/ fo all P In this case, the above expession, fo each choice of S, matches the claimed uppe bound in Theoem 16 The case s = 1 tuns out to be moe delicate Fo even, the optimal choice tuns out to be n P = n / fo P an edge of a pefect matching of the complete gaph with vetex set [], and othewise n P = 1 Fo odd, we have thee diffeent edge weights The gaph on [] whose edges consist of those pais with weight not equal to 1 consist of a disjoint union of a tiangle and a matching with 3/ edges The edges of the tiangle each have weight n 1/ log n 3/ and the edges of the matching each have weight n / log n 3/ It is staightfowad to check that these choices of weights give the claimed uppe bound in Theoem 16 Simila to the case = 3 and s = mentioned above, using Gallai s stuctue theoem, we obseve that, in any -coloing of the edges of the complete gaph on n vetices without a ainbow tiangle, the complete gaph can be edge-patitioned into subgaphs, each of which is a -coloed pefect gaph A simple agument then shows that thee is a vetex subset of at least n s / vetices which uses at most s colos, which veifies the lowe bound in Theoem 16 apat fom the logaithmic factos Impoving the lowe bound futhe to Theoem 16 is moe involved, using a weighted vesion of Ramsey s theoem and a caefully chosen induction agument to pove this The est of the pape is oganized as follows In the next section, we pove some basic popeties of lexicogaphic poduct coloings In Section 3, we give simple poofs of lowe and uppe bounds in the diection of Theoem 11 which match apat fom two logaithmic factos In ode to close the gap and obtain Theoem 11, in Section 4 we pove a weighted extension of Ramsey s theoem We complete the poof of Theoem 11 in Section 5 by establishing a tight lowe bound on the size of the lagest -coloed set of vetices in any Gallai 3-coloing of the complete gaph on n vetices The emaining sections ae devoted to the poof of Theoem 16 In Section 6, we pove the uppe bound fo Theoem 16 In Section 7, we give a simple poof of a lowe bound which matches Theoem 16 apat fom the logaithmic factos In Section 81, using the second moment method, we establish an 5
auxiliay lemma which gives a tight bound on the minimum possible numbe of nonzeo weights in a gaph with non-negative edge weights such that no set of s vetices contains sufficiently moe than the aveage weight of a subset of s vetices We give the lowe bound fo Theoem 16 in Section 8, which completes the poof of this theoem The poofs of some of the auxiliay lemmas which involve lengthy calculations ae given in the appendix All logaithms in this pape ae base, unless othewise indicated All coloings ae edge-coloings of complete gaphs, unless othewise indicated Fo the sake of claity of pesentation, we systematically omit floo and ceiling signs wheneve they ae not cucial We also do not make any seious attempt to optimize absolute constants in ou statements and poofs Lexicogaphic poduct coloings In this section, we will pove some simple esults about lexicogaphic poduct coloings Definition 1 These will be useful in constucting examples of -coloings that do not contain lage vetex sets that use at most s colos Fo such a lexicogaphic poduct coloing F 1 F with F 1 on m 1 vetices and F on m vetices, we will view the vetex set intechangeably as [m 1 m ] and [m 1 ] [m ] Fo the sake of bevity, we often efe to a lexicogaphic poduct coloing as simply a poduct coloing Definition 1 Fo F an edge-coloing of K n and S R a set of colos, we wite that a set Z of vetices is S-subchomatic in F if evey edge intenal to Z takes colos unde F only fom S g S,F When F and S ae clea fom context, we shall simply say that Z is subchomatic We will wite to be the size of the lagest subchomatic set of vetices If F is an edge-coloing constucted via a poduct of two othe coloings F 1, F, then the next lemma allows us to detemine g S,F in tems of g S,F1 and g S,F Lemma Fo any -coloings F 1, F of EK n1, EK n, espectively, and any set S R of colos, g S,F = g S,F1 g S,F, whee F = F 1 F Poof: Let Z a set of subchomatic vetices in F so Z V K n1 n be given We will fist show Z g S,F1 g S,F Take U [n 1 ] to be the set of u [n 1 ] such that thee is some v [n ] with u, v Z; that is, U is the subset of [n 1 ] that is used in Z Fo any u [n 1 ], take V u [n ] to be the set of v [n ] such that u, v Z, that is, V u is the subset of [n ] that is paied with u in Z By constuction, we have Z = u U {u} V u Theefoe, the set U must be subchomatic in F 1, as given distinct u 1, u U thee ae v 1, v so that u 1, v 1, u, v Z, and hence: Thus, U g S,F1 F 1 u 1, u = F u 1, v 1, u, v S Futhemoe, given u U we must have that V u is subchomatic in F, as given distinct v 1, v V u we have that F v 1, v = F u, v 1, u, v S 6
Theefoe, V u g S,F Hence, Z = V u = u U{u} V u g S,F = U g S,F g S,F1 g S,F u U u U Since Z was abitay, we get g S,F g S,F1 g S,F We now pove that g S,F g S,F1 g S,F, thus giving the desied esult: take U [n 1 ] a subchomatic set unde F 1 and V [n ] a subchomatic set unde F We claim that U V is subchomatic unde F Conside any distinct pais u 1, v 1, u, v U V If u 1 u then F u 1, v 1, u, v = F 1 u 1, u S, and if u 1 = u then F u 1, v 1, u, v = F v 1, v S If we choose U to have size g S,F1 and V to have size g S,F, we get g S,F1 g S,F = U V g S,F The next lemma states that the popety of being a Gallai coloing is peseved unde taking poduct coloings Lemma 3 If F 1, F ae Gallai -coloings of EK n1, EK n, espectively, then if F = F 1 F then F is a Gallai coloing Poof: Let any thee vetices u = u 1, u, v = v 1, v, w = w 1, w [n 1 ] [n ] be given We will show that they do not fom a ainbow tiangle unde F If u 1 = v 1 = w 1 then F u, v = F u, v, F u, w = F u, w, F v, w = F v, w and so u, v, w do not fom a ainbow tiangle by the assumption that F is a Gallai coloing If u 1, v 1, w 1 ae paiwise distinct then F u, v = F 1 u 1, v 1, F u, w = F 1 u 1, w 1, F v, w = F 1 v 1, w 1 and so u, v, w do not fom a ainbow tiangle by the assumption that F 1 is a Gallai coloing Othewise, exactly one pai of u 1, v 1, w 1 ae equal Assume without loss of geneality that u 1 = v 1, u 1 w 1, and v 1 w 1 We have: F u, w = F 1 u 1, w 1 = F 1 v 1, w 1 = F v, w, so again u, v, w do not fom a ainbow tiangle The following coollay states that we may take a poduct of any numbe of -coloings and the esult will be a Gallai coloing; since all -coloings ae Gallai coloings, it follows by induction fom the pevious lemma Coollay 4 If F 1,, F k ae -edge-coloings, then F 1 F k is a Gallai coloing 3 Simple bounds fo thee colos In this section we will demonstate simple uppe and lowe bounds in the case = 3 and s = We fist apply the techniques of the pevious section to demonstate a Gallai 3-coloing with no lage -coloed vetex set 7
Theoem 31 Thee is a Gallai 3-coloing on m vetices so that fo evey two colos S R, evey vetex set Z using colos fom S satisfies Z 4/9 + o1m 1/3 log m Poof: Take t = m 1/3 ; then t 3 is at least m Fo evey pai of colos P R, take FP to be a -coloing of EK t using colos fom P so that the lagest monochomatic clique has size at most log t Such a coloing exists by the lowe bound on Ramsey numbes poved by Edős and Szekees in [8] We define F a coloing on t 3 vetices by taking F = F {R1,R } F {R,R 3 } F {R1,R 3 } whee R 1, R, R 3 ae such that R = {R 1, R, R 3 } This is a Gallai coloing by Coollay 4 Fixing any set S of two colos, two of the above thee coloings have S-subchomatic sets of size at most log t, and the emaining one has size t, so the size of the lagest S-subchomatic set in F is at most t log t Since S is abitay, the size of the lagest S-subchomatic set fo any S R is at most t log t Resticting F to any m vetices will be a 3-Gallai coloing with no subchomatic set of size lage than t log t Note that since t = m 1/3, we have t = 1 + o1m 1/3, so t log t = 1 + o1m 1/3 logm 1/3 = 4/9 + o1m 1/3 log m We now poceed to pove that any Gallai 3-coloing on m vetices contains a subchomatic set on two colos of size at least m 1/3 Indeed, the next theoem is a stengthening of this statement, as it states that the geometic aveage ove S R of gs,f must be at least m 1/3 Since we have thee colos, will efe to them as ed, blue, and yellow Theoem 3 Fo any Gallai 3-coloing F on m vetices, S R g S,F m Poof: We poceed by induction on m to pove the theoem Define g to be the size of the lagest subchomatic set using only the colos blue and yellow, o to be the size of the lagest subchomatic set using only the colos ed and yellow, and p to be the size of the lagest subchomatic set using only the colos ed and blue A note on nomenclatue: g stands fo geen, as blue and yellow fom geen when mixed Similaly, o stands fo oange and p fo puple We wish to show that gop n If m = 1, then g = o = p = 1 and gop = m Othewise, m > 1 and by the stuctue theoem fo Gallai coloings thee is a non-tivial patition of the vetex set into pats V 1,, V t and a pai of colos Q R satisfying that fo any distinct i, j [t] thee is a q Q so that evey edge between V i and V j has colo q Take m i to be the size of V i Take g i to be the size of the lagest set using only the colos blue and yellow fom V i, o i to be the size of the lagest set using only the colos ed and yellow fom V i, and p i to be the size of the lagest set using only the colos ed and blue fom V i Without loss of geneality we assume that Q contains colos blue and yellow We have g = i g i Indeed, we may combine all the lagest sets using colos blue and yellow fom each V i to obtain a set of size i g i that only uses blue and yellow Futhemoe, o max i o i and p max i p i This gives: gop = i g i op i g i o i p i i m i = m, 8
whee the last inequality follows by the induction hypothesis applied to F esticted to V i Note that we use o max i o i, p max i p i It is on these inequalities that we will in the next sections gain multiple factos of log m; if, fo example, we find some set U [t] satisfying that fo each distinct i, j U the edges between V i, V j ae all yellow, then o i U o i If it wee the case that the o i, p i wee all paiwise equal, then we would get by the Edős-Szekees bound fo Ramsey numbes that op = Ωlog t max i o i p i ; this motivates the appoach in the next two sections, whee we handle the geneal case in which it may not be tue that the o i, p i ae all paiwise equal 4 A weighted Ramsey s theoem In this section we will pove a vesion of Ramsey s theoem that will apply to gaphs in which the weight of a vetex may depend on the colo of the clique that contains the vetex The next lemma is a convenient statement of a quantitative bound on the classical Ramsey s Theoem Lemma 41 In evey -coloing of the edges of K t, fo some k and l thee is a ed clique of ode k and a blue clique of ode l with kl 1 4 log t Poof: Take k to be the ode of the lagest ed clique and l to be the ode of the lagest blue clique We must have k + l t < Rk + 1, l + 1 k It is outine to check that this implies kl 1 4 log t Fo the est of this pape, let M := 16 The following lemma, which we call the weighted Ramsey s theoem, states that if vetex i contibutes weight α i to any ed clique in which it is contained and weight β i to any blue clique in which it is contained, then we may give a lowe bound fo the poduct of the sizes of the lagest weighted ed and blue cliques Lemma 4 Given a -coloing of the edges of a complete gaph on t vetices with t M and vetex weights α i, β i, take γ i = α i β i and γ = min i γ i Thee is a ed clique S and a blue clique U with β u γ 3 log t s S α s u U Poof: The poof will dyadically patition the vetices based on thei pai of weights α i, β i, and then apply the classical Edős-Szekees bound on Ramsey numbes in the fom of the pevious lemma That is, we will find a lage set of vetices A so that any two vetices in A have simila values fo α i and β i Applying Lemma 41 to this set will give the desied esult Take α = max i α i and β = max i β i If αβ γ 3 log t we may take S = {i} with α i = α and U = {j} with β j = β Othewise, αβ/γ < 1 3 log t Obseve that fo each i we have α i α, β i β, and α i β i γ This gives γ/β α i α and γ/α β i β Note we may patition [γ/β, α] into m 1 logαβ/γ+1 intevals I 1,, I m1 such that, within any inteval I i, we have supi i / infi i Similaly, we may 9
patition [γ/α, β] into m logαβ/γ + 1 intevals I 1,, I m with supi i / infi i By the pigeonhole pinciple thee must be some pai j, j such that, taking A := {i : α i I j, β i I j }, we have A t/m 1 m Applying the pevious lemma to A, we get that thee is a ed clique S of size k and a blue clique U of size l with kl 1 4 log t/m 1 m Note since t M we get m 1 m log 1 3 log t + 1 = log 1 16 log t t 1/4 Theefoe, we get 1 4 log t/m 1 m 1 4 log t 3/4 1 8 log t Take α A = min i A α i and β A = min i A β i Fo any i A, α i I j and hence α A α i / Similaly, fo any i A we have β A β i / Theefoe, fixing any i A, we get α A β A α i β i γ/4 Theefoe, α s β u α A β A = kα A lβ A klγ/4 γ 3 log t s S u U s S u U Since in the statement of the weighted Ramsey s theoem we take γ = min i α i β i, it povides good bounds when α i β i does not vay much between the vetices Theefoe, when we wish to use it in the upcoming sections, we will fist dyadically patition the vetices based on α i β i and then apply the lemma to each patition Note that we chose γ = min i α i β i We may hope to be able to use othe functions of α i, β i in this expession Howeve, it is not as obust as one may hope In paticula, we want to obseve that the function α i + β i will not yield an analogous theoem, as if we have many vetices of weight 0, 1 and colo all of the edges ed, then the lagest ed clique has size 0 and the lagest blue clique has size 1, but fo each i we have α i + β i = 1 Fotunately, using α i β i will suffice fo ou puposes 5 Tight lowe bound fo thee colos In this section we will show that any Gallai 3-coloing on m vetices has a -coloed set of size Ωm 1/3 log m This matches the uppe bound up to a constant facto We will efe to the thee edge colos as ed, blue, and yellow Fo the est of this section, fix an intege m N We emak that in this section thee is an inductive agument fo which it is impotant to note that m emains fixed thoughout Let c log Cn if 0 < n m 4/9 fn := c log m 4/9 log Cnm 4/9 if m 4/9 < n m 8/9 c 3 log 4 m 4/9 log Cnm 8/9 if m 8/9 < n m, whee D = 048, C = D8, and c = log C = D 16 /4 We will have a futhe discussion about f and its popeties shotly Fo now, simply note that fm = Ωlog 6 m We will pove the following theoem, Theoem 51 Fo any n [m], a Gallai coloing F on n vetices has eithe max S g S,F S g nfn S,F m 7/18 /8 o 10
Befoe we pove Theoem 51, we show how it implies the existence of a lage subchomatic set Theoem 5 Evey Gallai 3-coloing of EK m has a two coloed set of size Ωm 1/3 log m Poof: By Theoem 51, we have that eithe max S g S,F m 7/18 /8 Ωm 1/3 log m, o g S,F mfm = c 3 m log 4 m 4/9 log Cm 1/9 c 3 m 6 log 4 m 9 log m = 15 c 3 m log 6 m S As we have a lowe bound on the poduct of thee numbes, one of these numbes must be at least the cubed oot Hence, max S g S,F 5 cm 1/3 log m Ωm 1/3 log m, as desied We will now poceed with a futhe discussion about f We call 0, m 4/9 ], m 4/9, m 8/9 ], m 8/9, m] the intevals of f Note that on each inteval, fn = γ log δn fo some constants γ, δ whee m is viewed as a constant Intuitively, C is lage so that we avoid the ange of values in which log is pooly behaved, and c is small both so that we may assume n is lage and to make the tansitions between intevals easie f was chosen so that it satisfies cetain popeties, the moe inteesting of which we explicitly enumeate below All of these popeties ae fomalizations of the statement f does not gow too quickly Lemma 53 If m C, then the following statements hold about f fo any intege n with 1 < n m 1 Fo any α [ 1 n, 1], fαn αfn Fo any α 1, α, α 3 [ 1 n, 1] such that i α i = 1 we have, taking n i = α i n, nfn i n i fn i 8 log C nfn 3 Fo i 0 and m 7/18 j 1 we have f i log D j 51f i+ 8 7 j 4 Fo 1 τ n D 3 τ, we have fτ fn/ 5 Fo any α [ 1 n, 1 3], fαn 16αfn These popeties ae collectively efeed to as the facts about f and ae poved in Appendix A We now poceed with a poof of Theoem 51 Poof of Theoem 51: We poceed by induction on n Define g to be the size of the lagest set in F using only the colos blue and yellow, o to be the size of the lagest set in F using only the colos ed and yellow, and p to be the size of the lagest set in F using only the colos ed and blue We wish to show that eithe gop nfn o maxg, o, p m 7/18 /8 Ou base cases ae those n fo which fn 1, as fo these cases by Theoem 3 gop n nfn Since c = log C, any n < C is a base case If we ae not in a base case, we have n C Since F is a Gallai coloing, thee is a non-tivial patition V K n = V 1 V t with V 1 V t 1 such that thee is some -coloing χ of [t] such that fo evey distinct i, j [t] and u V i, v V j, the colo unde F of {u, v} is χi, j 11
Suppose without loss of geneality that χ only uses the colos blue and yellow The poof will split into thee cases Cases 1 and, Peliminay Discussion: These will be the cases in which V 1 has a substantial potion of the vetices Let U 1 = V 1, U denote the union of V j ove j 1 such that χ1, j is yellow, and U 3 denote the union of V j ove j 1 such that χ1, j is blue We have that U 1, U, U 3 is a non-tivial patition of V Let n i = U i Let α i = U i /n = n i /n fo i = 1,, 3, so α 1 + α + α 3 = 1 Fo i = 1,, 3, let F i be the coloing F esticted to U i Let g i be the size of the lagest subchomatic set in F i using only the colos blue and yellow, o i be the size of the lagest subchomatic set in F i using only the colos ed and yellow, and p i be the size of the lagest subchomatic set in F i using only the colos ed and blue Suppose without loss of geneality n n 3, so α 1 α 1 / and maxα 1, α 1/3 By the induction hypothesis, fo i = 1,, 3, we have that eithe one of g i, o i, p i is at least m 7/18 /8, in which case we may use g max i g i, o max i o i, p max i p i to complete the induction, o g i o i p i n i fn i Assume we ae in this latte case Since the U i ae connected only by yellow and blue edges, we may take the lagest subchomatic set using only yellow and blue fom each U i, giving g g 1 + g + g 3 in fact, equality holds Since U 1 and U ae connected with yellow edges, we may take the lagest subchomatic set using only ed and yellow fom both U 1 and U, o we may simply take the lagest such subchomatic set fom U 3, so we get o maxo 1 + o, o 3 Similaly, p maxp 1 + p 3, p Note We thus have gop g 1 op + g op + g 3 op g 1 o 1 + o p 1 + p 3 + g o 1 + o p + g 3 o 3 p 1 + p 3 gop g 1 o 1 + o p 1 + g o 1 + o p + g 3 o 3 p 3 = g 1 o p 1 + g o 1 p + 3 g i o i p i 3 g i o i p i g 1 o p 1 + g o 1 p g 1 o p g o 1 p 1 = g 1 o 1 p 1 g o p i=1 n 1 fn 1 n fn α 1 α n fn 1 fn, whee the second inequality is an instance of the aithmetic-geometic mean inequality Case 1: α 1, α log C 1/4 In this case, we have gop 3 g i o i p i α 1 α n fn 1 fn α 1 α nfn nfn/ log C i=1 i=1 8 log C nfn nfn i n i fn i, whee the second inequality is by the fist fact about f, the thid inequality is by substituting lowe bounds on α 1 and α, and the last inequality is by the second fact about f Hence, gop 3 g i o i p i + nfn i i=1 1 n i fn i nfn,
whee the last inequality is by the induction on hypothesis applied to U i fo i = 1,, 3 This completes this case Case : α 1 log C 1/4 α Befoe we poceed with this case, we pove a simple claim Claim 54 nfn + n 1 fn 1 n 1 fn > 0 Poof: Note nfn n 1 fn = 1 α 1 nfn Theefoe, n 1 fn 1 n 1 fn α 1 n 1 fn n 1 fn = α1nfn α 1 nfn = α 1 1 α 1 nfn, whee the fist inequality follows fom the fist fact about f Fom this we get nfn + n 1 fn 1 n 1 fn 1 α 1 nfn α 1 1 α 1 nfn = 1 α 1 nfn > 0 In this case we have α 1 1 α + α 3 1 α 1 log C 1/4 1/ and hence 3 gop g i o i p i α 1 α n fn 1 fn 8α 1 α nfn 4α nfn i=1 α + α 3 nfn = n n 1 fn n n 1 fn nfn + n 1 fn 1 n 1 fn = nfn n 1 fn 1 nfn i n i fn i, whee the second inequality is by both the fist fact about f applied to fn 1 and the fifth fact about f applied to fn, the thid inequality is by α 1 1/, and the second-to-last one is by the claim Hence, gop 3 g i o i p i + nfn i=1 3 n i fn i nfn, whee the last inequality is by the induction on hypothesis applied to U i fo i = 1,, 3 This completes this case Case 3: α 1 < log C 1/4 This is the spase case, when each pat is at most a log C 1/4 = D faction of the total Take n i = V i Take F i to be the coloing F esticted to V i Take g i to be the size of the lagest subchomatic set in F i using only the colos blue and yellow, o i to be the size of the lagest subchomatic set in F i using only the colos ed and yellow, and p i to be the size of the lagest subchomatic set in F i using only the colos ed and blue We eode the V i so that if i j then o i p i o j p j Take τ = logd n, so max i n i log C 1/4 n D n τ D n Define, fo i τ, I i := [ i, i+1 ] Take Φi = {j : n j I i } The Φi ae dyadically patitioning the indices; we will eventually use these patitions to constuct sets to which we will apply the weighted Ramsey s theoem Note that g = j g j, so we have gop = j g jop We now pesent the idea behind the agument fo the est of this case Fix i so that Φi has at least D 7 8 τ i elements and i lognm 7/18 we will show that most vetices v ae contained in V j i=1 13
as j vaies ove the Φi that have this popety We will define a weighted gaph whose vetices ae the indices and whose coloing is χ Given an index j its weight will be o j, p j If we find a yellow clique in χ then the sum of the o j in the clique gives a lowe bound on o, and similaly if we find a blue clique in χ then the sum of the p j in the clique gives a lowe bound on p We will apply the weighted Ramsey s theoem to half of the indices in Φi to the indices that ae lage than the median of Φi, to be pecise; fom this, we will be able to conclude that if j is an index smalle than the median, then op/o j p j D fn/fn j fo some lage constant D and so g j op D g j o j p j fn/fn j D n j fn We now poceed with the agument When we count, we wish to omit pats Φi that don t satisfy desied popeties; take B := {i τ : Φi D 7 8 τ i }, B := {i lognm 7/18 } Take B = B B We will show that a lage faction of the vetices ae not contained in V j fo j Φi whee i anges ove B i B j Φi n j i τ i+1 D 7 8 τ i = 4D 7 8 τ i τ i 8 4D 7 8 τ 1 1/8 1 τ+1/8 8D 1 1/8 1 τ 18D τ 56 D n n/4, whee the fouth inequality follows fom 1/8 1 + 1/16 Note if i g i m 7/18 /8 then we may complete the induction; assume this is not the case paticula, we get t m 7/18 /8 since g i 1 Theefoe, i B j Φi n j i B j Φi nm 7/18 tnm 7/18 n/4 In Hence, i B j Φi n j i B j Φi n j + j Φi n j n/ i B j Φi n j n/4 + n/4 n/ As a coollay we get i B Fo any fixed i τ such that i B, take β i to be the median of Φi if Φi has an even numbe of elements, take β i to be the lage of the two medians Conside {o j, p j : j Φi, j β i } By i B, this has at least D 7 8 τ i M elements ecall fom the weighted Ramsey s theoem that M = 16, so we get by applying the weighted Ramsey s theoem to this set that op o βi p βi log D 7 τ i 8 /3 Finally, obseve that eithe one of the o j, p j, g j is at least m 7/18 /8 in which case we may conclude the induction, o by the induction hypothesis we may assume o j p j g j n j fn j Theefoe, j Φi g j op j Φi j Φi:j β i g j o j p j log g j o βi p βi log D 7 8 τ i /3 D 7 8 τ i /3 j Φi:j β i g j o βi p βi log j Φi:j β i n j fn j log D 7 8 τ i /3 D 7 8 τ i /3 14
j Φi:j β i n j f log n j log D 7 8 τ log n j /3 j Φi:j β i 16n j f τ j Φi:j β i 8n j fn, whee the thid inequality is by o j p j o j p j fo j j, the fouth inequality is by the induction hypothesis applied to V j, the sixth inequality is by the thid fact about f, and the seventh inequality is by the fouth fact about f and noting τ D 3 n We now conside fo any set J Φi: This gives: n j i J j J j Φi j J n j j Φi n j Noting that {j Φi : j β i } Φi /: n j i+1 Φi J Φi Theefoe, 8n j fn 1 4 j Φi:j β i 8n j fn = fn n j j Φi j Φi gop j g j op i τ j Φi fn g j op We have thus concluded the induction i τ:i B j Φi i τ:i B j Φi g j op i τ:i B n j fn n = nfn fn j Φi We infomally efe to B in the above poof as lage if a lage faction of the vetices ae contained in a V j fo j Φi whee i anges ove B The case in which B was lage easily implied the desied esult In extending this esult in Section 8 to moe colos, the pimay difficulty is the following: when s is not, it is not obvious that thee is a lage s-coloed set as a esult of B being lage n j = 6 Uppe bound fo many colos In this section we will give asymptotically tight uppe bounds fo how lage of a subchomatic set must exist in an edge coloing on m vetices We will fist show how to constuct such coloings fom weighted gaphs with vetex set R, and then we will choose such gaphs to finish the constuction The next theoem states that if we have a weighted gaph on vetices with edge weights w P, then we can find a coloing F so that g S,F is, up to logaithmic factos, P S w P Lemma 61 Given a weighted gaph R, P on vetices with intege edge weights {w P } P P, taking m := P P w P, thee is a Gallai -coloing on m vetices so that fo any S R, the size of the lagest subchomatic set with colos in S is at most P P:P S w P P P: P S =1 log w P 15
Poof: We may define a Gallai -coloing on m vetices as follows: take P 1,, P k an abitay enumeation of P Fo each edge P, take F P to be a -coloing of EK wp using colos fom P so that the lagest monochomatic clique has ode at most log w P such a coloing exists by the Edős-Szekees bound fo Ramsey numbes [8] We define a coloing F on m vetices by F = F P1 F P F Pk F is a Gallai coloing by Coollay 4 Given any S R, note that g S,FP = w P if P S, as F P uses only colos fom P If P S = 1, then the lagest subchomatic set in F P using colos fom P S is at most log w P by choice of F P, so g S,FP log w P If P S = 0, then g S,FP = 1 as any two distinct vetices ae connected by an edge the colo of which is not in S Theefoe, g S,F = i g S,FPi P P:P S w P P P: P S =1 log w P The condition in the above lemma that the edge weights be integes is slightly cumbesome; we will now eliminate it Lemma 6 Fo any fixed intege 3, given a weighted gaph R, P on vetices with weights {w P } P P, taking m := P P w P, if m is an intege and each w P satisfies w P ω1, then thee is a Gallai -coloing on m vetices so that fo any S R, the size of the lagest subchomatic set is at most 1 + o1 P P:P S w P P P: P S =1 log w P Poof: Take w P = w P Since w P ω1, we get w P 1 + o1w P We may apply the pevious lemma to the w P to get an -Gallai coloing on P w P m vetices so that fo any S R the size of the lagest subchomatic set is at most P P:P S w P P P: P S =1 log w P 1 + o1 P P:P S w P P P: P S =1 log w P Restict this coloing to any m vetices; it is still a Gallai -coloing and fo any S R the size of the lagest subchomatic set is at most 1 + o1 P P:P S w P P P: P S =1 log w P Now, if we wish to obtain coloings without lage subchomatic sets, we need only constuct appopiate weighted gaphs Intuitively, we would like to minimize the numbe of edges in such a gaph while still being able to maintain that all the S R have appoximately the same value of P S w P, as evey edge ceates exta log factos This obsevation motivates the following bounds Theoem 63 Thee is a Gallai -coloing on m vetices so that fo any S R s the size of the lagest subchomatic set is at most 1 + o1m s / log c,s m, whee s s if s < 1, c,s = 1 if s = 1 and is even, + 3/ if s = 1 and is odd 16
Poof: If s < 1, we may apply the pevious lemma to a clique on vetices with edge weights m 1/ Any S R of size s has s intenal edges and s s edges intesecting it in one vetex By the pevious lemma, we may find a Gallai -coloing whee the size of the lagest subchomatic set is asymptotically at most: m s / log m 1/ s s m / s log m s s If s = 1 and is even, we may conside a pefect matching on vetices whee each edge has weight m / ; any subset of size 1 contains / 1 edges and thee is one edge with which it shaes exactly one vetex By the pevious lemma, we may find a Gallai -coloing whee the size of the lagest subchomatic set is asymptotically at most: m / 1// logm 1// m / 1// log m = m s / log m If s = 1 and is odd, we may conside a gaph fomed by taking the disjoint union of a tiangle on 3 vetices and a matching with 3/ edges The edges of the tiangle will each have weight w 1 := m 1/ log m 3/ and the edges of the matching will each have weight w := m / log m 3/ Note that the poduct of the weights is w 3 1 w 3/ = m Let S R of size s = 1 be given If the vetex not contained in S is pat of the tiangle then S contains 3/ edges of weight w and 1 edge of weight w 1 Futhemoe, thee ae two edges each of weight w 1 that S intesects in one vetex In the gaph obtained fom the pevious lemma the size of the lagest subchomatic set taking colos fom S is asymptotically at most: w 1 w 3/ log w 1 = m / log m 3/ logm 1/ log m 3/ m / log m 3/ log m = m s / log m +3/ If the vetex not contained in S is pat of the matching then S contains 5/ edges of weight w and 3 edges of weight w 1 Futhemoe, thee is one edge of weight w that intesects S in one vetex In the gaph obtained fom the pevious lemma the size of the lagest subchomatic set taking colos fom S is asymptotically at most: w 3 1w 5/ log w = m / log m 3/ logm / log m 3/ m / log m 3/ log m = m s / log m +3/ 7 Weak lowe bound fo many colos We now povide a simple lowe bound fo the lagest size of a subchomatic set in any -coloing of EK m that shows ou uppe bounds ae tight up to polylogaithmic factos; we show that any Gallai -coloing on m vetices contains a subchomatic set of size at least m s / The following is a common genealization of Hölde s inequality that we will find useful 17
Lemma 71 If S is a finite set of indices and fo each S S we have g S is a function mapping [t] to the non-negative eals, then S g S i g S i 1/ S S S i i S S Using the above lemma, we will pove a lowe bound on the poduct of the g S,F -coloing This will easily imply the desied lowe bound fo F a Gallai Theoem 7 Fo any Gallai -coloing F on m vetices, n s S R s g S,F Poof: Take g S = g S,F We poceed by induction on n If n = 1, then each g S is 1 as is thei poduct, while n s is also 1 If n > 1, we may find some pai of colos Q and some non-tivial patition of the vetices V 1,, V t such that fo each pai of distinct i, j in [t], thee is a q Q so that all of the edges between V i and V j have colo q Define, fo i [t], F i to be the estiction of F to V i Take g S,i := g S,Fi By induction, fo each i we have S g s S,i n i, whee n i = V i Note that if Q S then g S i g S,i, since we may combine the lagest subchomatic sets fom each F i Fo evey S we have g S max i g S,i, so i g S S S:Q S g 1/ s S,i S:Q S S:Q S i g S,i g 1/ s S S:Q S s g S i i S g 1/ s S,i S:Q S g 1/ s S,i s s i S:Q S g S = s n i = n s, whee the fist inequality is by g S i g S,i if Q S, the second inequality is by the peceding lemma and noting S = s, the thid inequality is by gs g S,i, and the fouth inequality is by the induction hypothesis Note that in poving this bound, if S Q = 1 we simply use g S g S,Fi As in the = 3, s = case, if we can find a set of indices V i1,, V ik so that between any two of them the edges use the colo contained in S Q, we may obtain a stonge lowe bound on g S We now conclude the agument Theoem 73 In any Gallai -coloing F on m vetices, thee is some S R s so that gs,f m s / Poof: By the pevious theoem, S R s g S,F must be some S with g S,F s m As this is a poduct ove s numbes, thee m s / s = m / s 18
8 Lowe bound fo many colos In this section we show that ou uppe bounds on sizes of subchomatic sets in Gallai coloings ae tight up to constant factos whee we view and s as constant 81 Discepancy lemma in edge-weighted gaphs The lemma in this subsection has the following fom: eithe a given weighted gaph has many edges of non-zeo weight o it has some set S of size s whose weight is significantly lage than aveage In the next subsection we will show how to educe the poblem of lowe bounding the size of the lagest subchomatic set in a Gallai -coloing to a poblem egading the numbe of non-zeo edges in a gaph that doesn t contain vetex subsets S whose weight is significantly lage than aveage, so this lemma will be useful Lemma 81 Given weights w P fo P R with wp 0, take w = P w P Take a 0 = if s < 1, a 0 = / if s = 1 and is even, and a 0 = + 3/ if s = 1 and is odd Eithe thee ae at least a 0 pais P with w P > 0 o thee is some S R of size s satisfying 1 s w P 1 + 4 w P S The poof of the above lemma uses elementay techniques along with the second moment method and is defeed to Appendix B 8 Poof of lowe bound fo many colos Let δ 0 = C 1 s 1 d = s 1 s = s s 1, 3 C = 3 d, 1 δ = 4 C, s 1 1 + 1 = C 1 d s 1, s 1 δ 1 = δ 1 0 + 1 1 c = δ/4 δ 1/d 1 1 + 1, d is an appopiately chosen scaling facto; why it is appopiate will become evident late C should be thought of as a lage constant, and δ, δ 0, δ 1, and c should be thought of as small constants We povide some bounds on the above; although we will not explicitly efeence these, they ae useful fo veifying vaious inequalities: 19
1 d, C 4 8, δ 8, δ 0 11 /4, δ 1 113, c 14 When we constucted the uppe bound via poduct coloings, thee was a weighted gaph namely the one used to constuct the coloing so that fo any S R we could appoximate the size of the lagest clique using colos fom S by the poduct of the weights of edges contained in S We wish to say that the stuctue of any Gallai coloing F can be appoximated this way Though this is not tue in geneal, the next theoem states that if it is not tue then S g S,F must be lage Take fo the est of this pape m 0 := 8 1 Theoem 8 If m m 0 then fo any Gallai coloing F on n m vetices, thee ae f 1, ɛ 0, P R, and, fo P P, weights wp [1, satisfying: 1 Fo evey S R s, gs,f P S P w P P P w P m ɛ n 3 S R s g s S,F nf 4 f log m Cɛ 5 Taking a to be the size of P, f c log m ad Fom the above theoem we will quickly be able to conclude Theoem 11 Note that if f is lage enough then by condition 3 we conclude that S R s g is lage and so some g S,F S,F is lage Othewise, by condition 4 we have an uppe bound on the size of ɛ, so by conditions 1 and the stuctue of the coloing is well-appoximated by the w P This latte case will allow us to apply ou wok on weighted gaphs fom the pevious subsection to get a lowe bound on a, and then we will apply condition 5 to as befoe conclude that some g S,F is lage Poof: We will wite g S fo g S,F We will take w P = 1 fo any P in R but not in P; this way, fo any T R, we have P T P w P = P T w P We poceed by induction on n Base Case: If n = 1, then we may take f = 1, ɛ = 0, and P = Letting a = P = 0, 1 Fo evey S R s, gs = 1 = P S w P P R w P = 1 = m ɛ n 0
3 S R s g = 1 = s S nf 4 f = 1 = log m Cɛ 5 f = 1 = c log m ad Peliminay Discussion: If n > 1, thee is some pai of colos Q = {Q 1, Q } and thee is a non-tivial patition V K n = V 1 V t with V 1 V t such that thee is some -coloing χ : t Q such that fo evey distinct i, j [t] and u V i, v V j, the colo unde F of {u, v} is χi, j which is in Q ɛ Given ɛ > 0, define f ɛ l := log m C logαm log m, whee α = l/n Note that we may ewite f ɛ l = Cɛ+C log α log m log m ; we will move between the two expessions feely Note also that f ɛ l is an inceasing function of l We will need some lemmas about f ɛ, all of which ae fomalizations of the statement f ɛ does not gow too quickly Lemma 83 The following statements hold about f ɛ fo evey choice of ɛ 0, m m 0, and 1 < n m 1 Fo any α [ 1 n, 1], f ɛ αn α 1/ s fɛ n In paticula, f ɛ αn αf ɛ n Fo any α 1, α, α 3 [ 1 n, 1] with α 1 + α + α 3 = 1, taking n i = α i n, nf ɛ n i n i f ɛ n i + 3log 3/4 mnf ɛ n 3 Fo i 0 and m δ j 1 we have f ɛ i log / s log 1/4 m j 56 fɛ i+j 4 Fo any α log 1 m, f ɛ αn f ɛ n/ We will efe to the above collectively as the facts about f ɛ ; we pove them in Appendix C The poof will split into fou cases Cases 1 and, Peliminay Discussion: Fo these cases, a simple numeical claim will be useful Claim 84 Fo positive eals a, b with a 1, 1 + a b 1 + ab/ Poof: Since 0 a 1 we have 1 + a e a/ Then 1 + a b e ab/ 1 + ab/ Cases 1 and will be those cases in which V 1 is lage 1