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Dscrete Mathematcs 309 (009) 4590 4595 Contents lsts avalable at ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Stablty of the path path Ramsey number András Gyárfás a, Gábor N. Sárközy a,b,, Endre Szemeréd c,d a Computer and Automaton Research Insttute, Hungaran Academy of Scences, Budapest, P.O. Box 63, Budapest, H-1518, Hungary b Computer Scence Department, Worcester Polytechnc Insttute, Worcester, MA, 01609, USA c Computer Scence Department, Rutgers Unversty, New Brunswck, NJ, 08903, USA d Insttute for Advanced Study, Prnceton, NJ, 08540, USA a r t c l e n f o a b s t r a c t Artcle hstory: Receved 18 August 008 Receved n revsed form 15 February 009 Accepted 18 February 009 Avalable onlne 0 March 009 Here we prove a stablty verson of a Ramsey-type Theorem for paths. Thus n any -colorng of the edges of the complete graph K n we can ether fnd a monochromatc path substantally longer than n/3, or the colorng s close to the extremal colorng. 009 Elsever B.V. All rghts reserved. Keywords: Ramsey theory Stablty Path 1. Introducton The vertex-set and the edge-set of the graph G s denoted by V(G) and E(G). K n s the complete graph on n vertces, K r+1 (t) s the complete (r +1)-partte graph where each class contans t vertces and K (t) = K(t, t) s the complete bpartte graph between two vertex classes of sze t. We denote by (A, B, E) a bpartte graph G = (V, E), where V = A + B, and E A B. For a graph G and a subset U of ts vertces, G U s the restrcton of G to U. The set of neghbors of v V s N(v). Hence the sze of N(v) s N(v) = deg(v) = deg G (v), the degree of v. The mnmum degree s denoted by δ(g) and the maxmum degree by (G) n a graph G. When A, B are subsets of V(G), we denote by e(a, B) the number of edges of G wth one endpont n A and the other n B. In partcular, we wrte deg(v, U) = e({v}, U) for the number of edges from v to U. A graph G n on n vertces s γ -dense f t has at least γ ( ) n edges. A bpartte graph G(k, l) s γ -dense f t contans at least γ kl edges. For graphs G 1, G,..., G r, the Ramsey number R(G 1, G,..., G r ) s the smallest postve nteger n such that f the edges of a complete graph K n are parttoned nto r dsjont color classes gvng r graphs H 1, H,..., H r, then at least one H (1 r) has a subgraph somorphc to G. The exstence of such a postve nteger s guaranteed by Ramsey s classcal result [1]. The number R(G 1, G,..., G r ) s called the Ramsey number for the graphs G 1, G,..., G r. There s very lttle known about R(G 1, G,..., G r ) even for very specal graphs (see e.g. [4] or [11]). For r = a theorem of Gerencsér and Gyárfás [3] states that 3n R(P n, P n ) =. In ths paper we prove a stablty verson of ths theorem. Snce ths s what we needed n a recent applcaton [6], actually we prove the result n a slghtly more general context; we work wth -edge multcolorngs (G 1, G ) of a graph G. Here (1) Correspondng author at: Computer Scence Department, Worcester Polytechnc Insttute, Worcester, MA, 01609, USA. E-mal addresses: gyarfas@sztak.hu (A. Gyárfás), gsarkozy@cs.wp.edu (G.N. Sárközy), szemered@cs.rutgers.edu (E. Szemeréd). 001-365X/$ see front matter 009 Elsever B.V. All rghts reserved. do:10.1016/j.dsc.009.0.05
A. Gyárfás et al. / Dscrete Mathematcs 309 (009) 4590 4595 4591 multcolorng means that the edges can receve more than one color,.e. the graphs G are not necessarly edge dsjont. The subgraph colored wth color only s denoted by G,.e. G 1 = G 1 \ G, G = G \ G 1. In order to state the theorem we need to defne a relaxed verson of the extremal colorng for (1). Extremal Colorng (wth parameter α): There exsts a partton V(G) = A B such that A (/3 α) V(G), B (1/3 α) V(G). The graph G 1 A s (1 α)-dense and the bpartte graph G A B s (1 α)-dense. (Note that we have no restrcton on the colorng nsde the smaller set.) Then the followng stablty verson of the Gerencsér Gyárfás Theorem clams that we can ether fnd a monochromatc path substantally longer than n/3, or the colorng s close to the extremal colorng. Theorem 1.1. For every α > 0 there exsts a postve real η (0 < η α 1 where means suffcently smaller) and a postve nteger n 0 such that for every n n 0 the followng holds: f the edges of the complete graph K n are -multcolored then we have one of the followng two cases. Case 1: K n contans a monochromatc path P of length at least ( 3 + η)n. Case : Ths s an Extremal Colorng (EC) wth parameter α. We remark that whle for some classcal densty results the correspondng stablty versons are well-known (see [1]), stablty questons n Ramsey problems only emerged recently (see [5,7,10]).. Tools Theorem 1.1 can also be proved from the Regularty Lemma [13], however, here we use a more elementary approach usng only the Kővár Sós Turán bound [8]. Ths s part of a new drecton to de-regularze some proofs, namely to replace the Regularty Lemma wth more elementary classcal extremal graph theoretc results such as the Kővár Sós Turán bound (see e.g. [9]). Lemma.1 (Theorem 3.1 on page 38 n [1]). There s an absolute constant > 0 such that f 0 < ɛ < 1/r and we have a graph G wth E(G) (1 1r ) n + ɛ then G contans a K r+1 (t), where log n t =. r log 1/ɛ For r = 1 ths s essentally the Kővár Sós Turán bound [8] and for general r ths was proved by Bollobás, Erdős and Smonovts []. Here we wll use the result only for r = 1. 3. Outlne of the proof We wll need the followng defnton. Gven a graph G and a postve nteger k, we say that a subset W of the vertex set V(G) s k-well-connected f for any two vertces u, v W there are at least k nternally vertex dsjont paths of length at most three connectng u and v n G (note that these paths mght leave W ). We wll use ths defnton wth k = ηn, n ths case we just say shortly that W s well-connected. We wll follow a smlar outlne as n applcatons of the Regularty Lemma. However, a regular par wll be replaced wth a complete balanced bpartte graph K(t, t) wth t c log n for some constant c (thus the sze of the par s somewhat smaller but ths s stll good enough for our purposes). Then a monochromatc connected matchng n the reduced graph (the usual tool n these types of proofs usng the Regularty Lemma) wll be replaced wth a set of vertex dsjont monochromatc complete balanced bpartte graphs K (t, t ), 1 s wth t c log n, 1 s for some constant c. Moreover, these bpartte graphs are all contaned n a set W that s well-connected n ths color. Let us call a set of bpartte graphs lke ths a monochromatc well-connected complete balanced bpartte graph cover (we are coverng vertces here). The sze of ths cover s the total number of vertces n the unon of these complete bpartte graphs. Then Theorem 1.1 wll follow from the followng lemma. Lemma 3.1. For every α > 0 there exst a postve real η (0 < η α 1 where means suffcently smaller) and a postve nteger n 0 such that for every n n 0 the followng holds: f the edges of the complete graph K n are -multcolored then we have one of the followng two cases. Case 1: K n contans a monochromatc well-connected complete balanced bpartte graph cover of sze at least ( 3 + η)n. Case : Ths s an Extremal Colorng (EC) wth parameter α.
459 A. Gyárfás et al. / Dscrete Mathematcs 309 (009) 4590 4595 Indeed, let us assume that we have Case 1 n ths Lemma. Denote the two color classes of K (t, t ) by V 1 and V for 1 s. Snce ths cover s nsde the same set W that s well-connected n ths color (say red), we can fnd a red connectng path P 1 of length at most 3 from an arbtrary vertex of V 1 to an arbtrary vertex of V 1. Smlarly we can fnd a red connectng path P of length at most 3 from a vertex of V to a vertex of V 3 1 that s vertex dsjont from P 1. We contnue n ths fashon untl we fnd a red connectng path P s 1 of length at most 3 from a vertex of V s 1 to a vertex of V s 1 that s vertex dsjont from each of the connectng paths P, 1 s, constructed so far. Furthermore, we can also guarantee that n ths connectng process from any V j, 1 j, 1 s we never use up more than η Vj vertces. Indeed, then durng the whole process the total number of forbdden vertces s at most 4s η 4n cη log n ηn, (f n s suffcently large) and thus we can always select the next connectng path that s vertex dsjont from the ones constructed so far. We remove the nternal vertces of these connectng paths P from the complete balanced bpartte graphs K (t, t ). By dong ths we may create some dscrepances n the cardnaltes of the two color classes. We remove some addtonal vertces to assure that now we have the same number t ( (1 η)t ) of vertces left n both color classes. Now we can connect the endpont of P 1 n V 1 and the endpont of P n V by a red Hamltonan path Q n the remander of K (t, t ) (usng the fact that ths s a balanced complete bpartte graph). Puttng together the connectng paths P and these Hamltonan paths Q we get a red path P of length at least ( 3 + η ) ( ) 3 η n 3 + η n. Actually, n the applcaton [6] we needed a slghtly stronger statement, namely that for each vertex of P we have at least (c/) log n choces, but ths s a straghtforward corollary of the proof. Corollary 3.. In Case 1 of Theorem 1.1, n the process of fndng P, for each vertex of the path P we have at least c 1 log n choces for some constant c 1. 4. Monochromatc well-connected components In ths secton we show that n any -multcolorng of K n there s a large set W and a color such that W s k-well-connected n ths color. Lemma 4.1. For every nteger k and for every -multcolored K n there exst W V(K n ) and a color (say color 1) such that W n 8k and W s k-well-connected n the color 1 subgraph of K n. Proof. Assume that a -multcolorng s gven on K n - n fact t s enough to prove the lemma for a colorng obtaned by gnorng one of the colors of every -colored edge. A par u, v V(K n ) s bad for color 1 f there are no k nternally vertex dsjont paths of length at most three from u to v all monochromatc n color 1. Let m be the maxmum number of vertex dsjont bad pars for color 1. If m < k then deletng m < 4k vertces of a maxmum matchng of bad pars for color 1, we have a set W of more than n 4k vertces that s k-well-connected for color 1 and the proof s fnshed (wth 4k to spare). Note that deletng these m vertces wll not destroy the short connectng paths between the remanng pars of vertces, snce these paths are allowed to leave W, so they may use the deleted vertces. Otherwse select a matchng {u 1 v 1,..., u k v k } of k bad pars for color 1. For t [k],, j [1, ] defne A(t,, j) as the set of vertces adjacent to u t n color and to v t n color j. From the defnton of bad pars A(t, 1, 1) < k. For the same reason usng Köng s theorem all edges of color 1 n the bpartte graph [A(t, 1, ), A(t,, 1)] can be met by a set S t of less than k vertces. Set B t = A(t, 1, ) \ S t, C t = A(t,, 1) \ S t. Observe that between B t and C t we have a complete bpartte graph n color (f both are non-empty). A set B t (C t ) s called small f t has less than k elements. Defne H t as the unon of A(t, 1, 1) S t and the small sets B t, C t (that s, we nclude B t or C t only when the set s small). Observe that H t < 6k. Consder the hypergraph H wth vertex set X = V(K n ) \ ( t [k] {u t, v t }) and edge set X H t for t [k]. Let M be the set of vertces of H wth degree at least k/. Then k 1k = k 6k H t = x X t=1 deg(x) H x M deg(x) M (k/) H mplyng that 4k M. Therefore X \ M n 4k 4k = n 8k. We clam that W = X \ M s k-well-connected n color and that wll fnsh the proof. To prove the clam, consder a par of vertces x, y W. From the defnton of W, x, y are both n less than k/ edges of H. Thus x, y are both covered by at least k sets n the form Y t = (,j [1,] A(t,, j)) \ H t where t [k]. Therefore n every Y t the par x, y can be connected n color ether by a path of length two through u t (f x, y A(t,, ) A(t,, 1)) or through v t (f x, y A(t,, ) A(t, 1, )) or through a path of length three (f one of x, y s n A(t,, 1) and the other s n A(t, 1, ).
A. Gyárfás et al. / Dscrete Mathematcs 309 (009) 4590 4595 4593 It s easy to see usng that there are at least k choces for t, B t = A(t, 1, ) \ S t k, C t = A(t,, 1) \ S t k and that we have a complete bpartte graph between B t and C t n color that there are at least k nternally edge dsjont paths of length at most three n color connectng x, y. 5. Proof of Lemma 3.1 Assume that we have an arbtrary -multcolorng (red/blue) of K n. We shall assume that n s suffcently large and use the followng man parameters 0 η α 1, where a b means that a s suffcently small compared to b. In order to present the results transparently we do not compute the actual dependences, although t could be done. We wll use the constant ( η ) 8 c = η 3, log 1 η where s from Lemma.1. Let us apply Lemma 4.1 wth k = ηn to fnd a W V(K n ) and a color (say red) such that W (1 8η)n and W s ηn-well-connected (or shortly, well-connected) n the red subgraph of K n. Put R = K n \ W, then we have R 8ηn. From now on we wll work nsde W. We may assume that nsde W the red densty s at least η, snce otherwse we can swtch colors as the blue-only subgraph s almost complete. Thus we can apply Lemma.1 wth r = 1 and ɛ = η to the red subgraph nsde W to fnd a red complete balanced bpartte subgraph K 1 (t 1, t 1 ) n W wth t 1 = log n log 1 η (for smplcty we assume that ths s an nteger). We remove ths K 1 (t 1, t 1 ) from W and n the remander of W teratvely we fnd red complete balanced bpartte graphs K(t 1, t 1 ) whle we can. Suppose that we found ths way the red well-connected complete balanced bpartte graph cover M 1 = (K 1 (t 1, t 1 ), K (t 1, t 1 ),..., K s1 (t 1, t 1 )) for some postve nteger s 1. If ths red cover M 1 has sze M 1 (/3+η)n, then we are done, we have Case 1 n Lemma 3.1. Otherwse we wll show that we can ether ncrease the sze of ths red cover by an η /-fracton, or we can fnd drectly a monochromatc well-connected complete bpartte graph cover of sze at least (/3 + η)n unless we are n the Extremal Colorng (Case ), as desred. We know that at least we have M 1 η 4 n, snce otherwse n the remander of W the red densty s stll at least η/, and we can stll apply Lemma.1 n the remander to fnd a red K(t 1, t 1 ). Let K (t 1, t 1 ) = (V 1, V ), 1 s 1. Denote From we have V 1 = s 1 =1 V 1 + V < V 3 > V 1, V = s 1 =1 ( ) 3 + η n, ( ) 1 3 30η n. V and V 3 = W \ (V 1 V ). Furthermore, snce n V 3 we cannot pck another red complete balanced bpartte subgraph K(t 1, t 1 ), by Lemma.1 V 3 s (1 η)-dense n the blue-only subgraph. Next let us look at the bpartte graphs (V 1, V 3 ) and (V, V 3 ). We wll show that ether one of them s (1 η)-dense n blue-only or we can ncrease our red cover M 1. Indeed, assume frst the followng: () There s a subcover of M 1 M 1 = (K 1 (t 1, t 1 ), K (t 1, t 1 ),..., K s 1 (t 1, t 1 )) wth 1 s 1 s 1 such that f we denote V = 1 s 1 j=1 V j 1, V = s 1 j=1 V j we have V = 1 V η V 1 = η V, and the bpartte graphs (V j, 1 V 3) and (V j, V 3) are both η-dense n red for every 1 j s 1. () (3) (4)
4594 A. Gyárfás et al. / Dscrete Mathematcs 309 (009) 4590 4595 Consder the bpartte graph (V j, 1 V 3) for some 1 j s 1. Snce ths s η-dense n red, there must be at least η V 3 / vertces n V 3 for whch the red degree n V j 1 s at least η V j 1 / = ηt 1/. Indeed, otherwse the total number of red edges would be less than η V 3 V j + η 1 V 3 V j = 1 η V 3 V j, 1 a contradcton wth the fact that (V j 1, V 3) s η-dense n red. Consder all the red neghborhoods of these vertces n V j 1. Snce there can be at most t 1 = n log 1 η such neghborhoods, by averagng and usng () and (4) there must be a red neghborhood that appears for at least η V 3 n log 1 η η 1 8 n log η 1 η t 1 such vertces of V 3. Ths means that we can fnd a red complete balanced bpartte graph K(t, t ) n the red bpartte graph (V j, 1 V 3), where t = ηt 1 /. We can proceed smlarly for the red bpartte graph (V j, V 3). Thus the red complete balanced bpartte graph K j (t 1, t 1 ) can be replaced wth three red complete balanced bpartte graphs. We can proceed smlarly for all K j (t 1, t 1 ), 1 j s 1. Ths way we obtan a new red well-connected complete bpartte graph cover M = (K 1 (t (), 1 t() ), 1 K (t (), t() ),..., K s (t () s, t () s )) such that ) M (1 + η M 1, and t 1 t () t = ηt 1 / for every 1 s. Iteratng ths process (j 1) tmes we get a new cover M j = (K 1 (t (j) 1, t(j) 1 ), K (t (j), t(j) s j ),..., K s j (t (j) such that ) j 1 M j (1 + η M1, and t 1 t (j) t j = (η/) j 1 t 1 for every 1 s j., t (j) s j )) Ths and (3) mply that f we could terate ths process 8/η 3 tmes, then we would get a red complete balanced bpartte graph cover of sze at least (/3 + η)n, where for each K (t, t ) n the cover we would stll have t ( η ) 8 η 3 t 1 = ( η ) 8 η 3 log 1 η log n = c log n and thus we would have Case 1 n Lemma 3.1. Indeed, usng (3) n each teraton we ncrease the sze of the cover by η M 1 η3 8 n, and thus n at most 8/η 3 teratons we get a cover of sze at least (/3 + η)n. We may assume that ths s not the case and after (j 1) teratons for some j < 8/η 3 we get a cover M j as above for whch () does not hold. In ths case we wll show that we can fnd drectly a monochromatc well-connected complete bpartte graph cover of sze at least (/3 + η)n unless we are n the Extremal Colorng (Case ). For smplcty we stll use the notaton V 1, V, V 3 for M j just as for M 1. Snce () does not hold for M j, for most of the complete bpartte graphs K (t (j), t (j) ) = (V, 1 V ) (namely for complete bpartte graphs coverng at least a (1 η)-fracton of the total sze of M j) the red densty n one of the bpartte graphs (V, 1 V 3) and (V, V 3) s less than η. By renamng we may assume that ths s always the bpartte graph (V, V 3). Ths means that ndeed (V, V 3 ) s (1 η)-dense n blue-only, as we wanted. However, ths mples that ( ) 1 V 3 < 3 + α3 n, snce otherwse usng () we can easly fnd a blue well-connected complete balanced bpartte graph cover of sze at least (/3+η)n by teratvely applyng Lemma.1 n blue, frst n the bpartte graph (V, V 3 ) whle we can, and then contnung
A. Gyárfás et al. / Dscrete Mathematcs 309 (009) 4590 4595 4595 nsde V 3. Smlarly f the blue densty n the bpartte graph (V 1, V ) s at least α 3, then agan we can fnd a blue wellconnected complete balanced bpartte graph cover of sze at least (/3 + η)n by teratvely applyng Lemma.1 n blue, frst n the bpartte graph (V 1, V ), then n the bpartte graph (V, V 3 ), and fnally nsde V 3. Thus we may assume that the bpartte graph (V 1, V ) s (1 α 3 )-dense n red-only. Makng progress towards the Extremal Colorng, next let us look at the red densty nsde V. Assume frst that ths densty s at least α. Smlarly as above ths mples that the bpartte graph (V 1, V 3 ) s (1 α )-dense n blue-only, snce otherwse we can fnd agan a red cover of sze at least (/3+η)n. Thus the bpartte graph (V 1 V, V 3 ) s (1 α )-dense n blue-only. Ths n turn mples that V 1 V s (1 α)-dense n red-only snce otherwse we can fnd agan a blue cover of sze at least (/3 + η)n. Ths gves us the Extremal Colorng where A = V 1 V, B = V 3, G 1 s red and G s blue (actually here n a somewhat stronger form snce nsde B most of the edges are blue-only as well). Hence we may assume that V s (1 α )-dense n blue-only. Ths mples agan that (V 1, V 3 ) s (1 α)-dense n red-only and ths gves us the Extremal Colorng agan where A = V V 3, B = V 1, G 1 s blue and G s red. Ths fnshes the proof of Lemma 3.1. Acknowledgements The frst author s research s supported n part by OTKA Grant No. K683. The second author s research was supported n part by the Natonal Scence Foundaton under Grant No. DMS-0456401, by OTKA Grant No. K683 and by a Janos Bolya Research Scholarshp. The thrd author s research was supported n part by the Ellentuck Fund. References [1] B. Bollobás, Extremal Graph Theory, Academc Press, London, 1978. [] B. Bollobás, P. Erdős, M. Smonovts, On the structure of edge graphs II, J. London Math. Soc. 1 () (1976) 19 4. [3] L. Gerencsér, A. Gyárfás, On Ramsey-type problems, Ann. Unv. Sc. Budapest Eötvös Sect. Math. 10 (1967) 167 170. [4] R.L. Graham, B.L. Rothschld, J.H. Spencer, Ramsey Theory, John Wley & Sons, New York, 1990. [5] A. Gyárfás, M. Rusznkó, G.N. Sárközy, E. Szemeréd, Three-color Ramsey numbers for paths, Combnatorca 7 (007) 35 69. [6] A. Gyárfás, G.N. Sárközy, E. Szemeréd, Monochromatc Hamltonan 3-tght Berge cycles n -colored 4-unform hypergraphs (submtted for publcaton). [7] Y. Kohayakawa, M. Smonovts, J. Skokan, The 3-colored Ramsey number of odd cycles, manuscrpt. [8] P. Kővár, V.T. Sós, P. Turán, On a problem of Zarankewcz, Colloq. Math. 3 (1954) 50 57. [9] I. Levtt, G.N. Sárközy, E. Szemeréd, How to avod usng the regularty lemma; Pósa s conjecture revsted, Dscrete Math. (n press). [10] V. Nkforov, R.H. Schelp, Cycles and stablty, J. Combn. Theory Ser. B 98 (008) 69 84. [11] S.P. Radzszowsk, Small Ramsey numbers, Electron. J. Combn. (00) DS1. [1] F.P. Ramsey, On a problem of formal logc, Proc. London Math. Soc., nd Ser. 30 (1930) 64 86. [13] E. Szemeréd, Regular parttons of graphs, n: Colloques Internatonaux C.N.R.S. N o 60 - Problèmes Combnatores et Théore des Graphes, Orsay, 1976, pp. 399 401.