On Schelp's problem for three odd long cycles

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Abstract

We show that for every η>0 there exists n0 such that for every odd nn0 each 3-colouring of edges of a graph G with (4+η)n and minimum degree larger than (7/2+2η)n leads to a monochromatic cycle of length n. This result is, up to η terms, best possible.

Introduction

We use Rado's notation and write G(H1,,Hr) whenever for each colouring of the edges of G with r colours there exists i=1,2,,r, such that some copy of Hi in G has all its edges coloured by the ith colour. The smallest number N such that KN(H1,,Hr) is called the Ramsey number for graphs H1,,Hr and denoted R(H1,,Hr). Schelp observed that quite often G(H1,,Hr) for G with R(H1,,Hr) vertices provided only that the minimum degree of G is large enough. He proved such a result for a pair of paths [11] and stated an appropriate conjecture for the minimum density for which it is true for a pair of cycles (see [9] and [12]). The conjecture has been proved to ‘asymptotically’ hold by Benevides et al. [1] and Gyárfás and Sárközy [6] (see also White [13] for an ‘extremal’ version of this problem).

Schelp's problem for a triple of cycles or paths seems to be much harder to settle. Let us recall that the asymptotic value of the Ramsey number for three long odd cycles was first found by Łuczak [10]. Then, it was used by Kohayakawa, Simonovits, and Skokan [8] to show that for sufficiently large odd n we have R(Cn,Cn,Cn)=4n3, confirming an old conjecture of Bondy and Erdős [3]. Our goal is to show the following theorem.

Theorem 1

For every η>0 there exists n0 such that for every odd nn0 and every graph G on (4+η)n vertices with δ(G)(7/2+2η)n each colouring of the edges of G with three colours leads to a monochromatic cycle of length n.

Let us observe that the above result is ‘asymptotically’ best possible. Indeed, for an even n the complete balanced bipartite graph on n vertices is n/2-regular, so there exists a graph G with 4n vertices with minimum degree 7n/2 whose edges can be decomposed into three bipartite graphs.

In the paper we use approach introduced in [10] based on the observation that, due to Szemerédi's Regularity Lemma, the existence of a long monochromatic cycle follows from the fact that the ‘reduced graph’ contains a non-bipartite monochromatic component with a large matching (for a formal exposition of the method see [5]). This is also true for graphs with large minimum degree and one can mimic the argument from [1] to show that Theorem 1 is a direct consequence of the following result.

Lemma 2

For every ξ>0 there exists k0 such that for every kk0 and every graph G on (4+ξ)k vertices with δ(G)(7/2+ξ)k the following holds. For each colouring of the edges of G with three colours there exists a non-bipartite monochromatic component which contains a matching saturating at least (1+ξ/9)k vertices.

The idea of the proof of Lemma 2 is to reduce the three-colour problem to a two-colour problem, which, in turn, is solved by using a number of “one-colour” results, i.e. facts on the existence of matchings contained in one components in graphs. Thus, the structure of the paper goes as follows. We start with some results on components and matchings we shall use later on. Then we state Lemma 8, crucial for our argument, which reduces the problem to two colours, and show how it implies Lemma 2 and so also Theorem 1. The next section is devoted to a long and technical proof of Lemma 8. We conclude the paper with some open problems and remarks.

Section snippets

Components and matchings

Let us start with the following fact on the size of the largest monochromatic components in dense graphs whose edges are coloured with three colours.

Lemma 3

Let G=(V,E) be a graph on 4k vertices with δ(G)>7k/2. Then each colouring of the edges of G with three colours leads to a monochromatic component of size at least 2k.

Proof

Let F1=(V1,E1) be the largest monochromatic component of G, say, in the first colour, and let |V1|<2k. Consider the largest monochromatic bipartite component F2=(V2,E2) in the subgraph

Reduction to two colours

The key ingredient of our argument is a somewhat technical result, which reduces our problem to two colours. In a way one can consider it as a 2-colour version of 1-colour Lemma 6 from the last section. For sake of simplifying the notation let F=F(N;β1,β2,β3) denote the graph Hˆ(N(1β1β2β3),2β1N,2β2N,2β3N). Moreover, here and below we shall omit all floors and ceilings and always assume that constants are chosen in such a way that the numbers which denote the number of vertices and degrees of

Proof of Lemma 8

Let us recall that F=F(N;β1,β2,β3) is a graph obtained from the complete graph on (1+β1+β2+β3)N vertices from which we have removed edges lying in three disjoint subsets Z1, Z2, Z3 of sizes 2β1N, 2β2N, 2β3N respectively. We also set α=β1+β2+β3, Z=Z1Z2Z3, and V0=VZ. In this chapter we always assume that β1β2β31/4 and α1/2, which imply that |Z|N and so |V0|N/2. Moreover, we consider a graph GF which is b-dense in G for some bN/4N. Note that from the definition of b-density, if G is b

Final remarks and open problems

As we have already mentioned Kohayakawa, Skokan and Simonovits [8] proved that there exists n0 such that for every odd nn0 we have R(Cn,Cn,Cn)=4n3. Thus, one can expect that the following to hold.

Conjecture

There exists n0 such that for every odd n, nn0, and every graph G on 4n3 vertices with minimum degree larger than 7n/23 each 3-colouring of the edges of G leads to a monochromatic cycle of length n.

One can also consider the case of triples of even cycles (which is very similar to the case of

Acknowledgement

We wish to thank an anonymous referee for his/her valuable comments and suggestions.

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The first author partially supported by National Science Centre, Poland, grant 2017/27/B/ST1/00873.

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