13 research outputs found

    Maximum number of triangle-free edge colourings with five and six colours

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    Let k ≥ 3 and r ≥ 2 be natural numbers. For a graph G, let F(G, k, r) denote the number of colourings of the edges of G with colours 1,…, r such that, for every colour c ∈ {1,…, r}, the edges of colour c contain no complete graph on k vertices Kk. Let F(n, k, r) denote the maximum of F(G, k, r) over all graphs G on n vertices. The problem of determining F(n, k, r) was first proposed by Erdős and Rothschild in 1974, and has so far been solved only for r = 2; 3, and a small number of other cases. In this paper we consider the question for the cases k = 3 and r = 5 or r = 6. We almost exactly determine the value F(n, 3, 6) and approximately determine the value F(n, 3, 5) for large values of n. We also characterise all extremal graphs for r = 6 and prove a stability result for r = 5

    The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques

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    Let k:=(k1,,ks)\mathbf{k} := (k_1,\dots,k_s) be a sequence of natural numbers. For a graph GG, let F(G;k)F(G;\mathbf{k}) denote the number of colourings of the edges of GG with colours 1,,s1,\dots,s such that, for every c{1,,s}c \in \{1,\dots,s\}, the edges of colour cc contain no clique of order kck_c. Write F(n;k)F(n;\mathbf{k}) to denote the maximum of F(G;k)F(G;\mathbf{k}) over all graphs GG on nn vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every k\mathbf{k} and nn, there is a complete multipartite graph GG on nn vertices with F(G;k)=F(n;k)F(G;\mathbf{k}) = F(n;\mathbf{k}). Also, for every k\mathbf{k} we construct a finite optimisation problem whose maximum is equal to the limit of log2F(n;k)/(n2)\log_2 F(n;\mathbf{k})/{n\choose 2} as nn tends to infinity. Our final result is a stability theorem for complete multipartite graphs GG, describing the asymptotic structure of such GG with F(G;k)=F(n;k)2o(n2)F(G;\mathbf{k}) = F(n;\mathbf{k}) \cdot 2^{o(n^2)} in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So

    Integer colorings with forbidden rainbow sums

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    For a set of positive integers A[n]A \subseteq [n], an rr-coloring of AA is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of [n][n] with the maximum number of rainbow sum-free rr-colorings. We show that for r=3r=3, the interval [n][n] is optimal, while for r8r\geq8, the set [n/2,n][\lfloor n/2 \rfloor, n] is optimal. We also prove a stability theorem for r4r\geq4. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

    On two problems in Ramsey-Tur\'an theory

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    Alon, Balogh, Keevash and Sudakov proved that the (k1)(k-1)-partite Tur\'an graph maximizes the number of distinct rr-edge-colorings with no monochromatic KkK_k for all fixed kk and r=2,3r=2,3, among all nn-vertex graphs. In this paper, we determine this function asymptotically for r=2r=2 among nn-vertex graphs with sub-linear independence number. Somewhat surprisingly, unlike Alon-Balogh-Keevash-Sudakov's result, the extremal construction from Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with sub-linear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an nn-vertex KkK_k-free graph GG with α(G)=o(n)\alpha(G)=o(n). The extremal graphs have similar structure to the extremal graphs for the classical Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page

    Colouring set families without monochromatic k-chains

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    A coloured version of classic extremal problems dates back to Erd\H{o}s and Rothschild, who in 1974 asked which nn-vertex graph has the maximum number of 2-edge-colourings without monochromatic triangles. They conjectured that the answer is simply given by the largest triangle-free graph. Since then, this new class of coloured extremal problems has been extensively studied by various researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of Sperner's Theorem, the classic result in extremal set theory on the size of the largest antichain in the Boolean lattice, and Erd\H{o}s' extension to kk-chain-free families. Given a family F\mathcal{F} of subsets of [n][n], we define an (r,k)(r,k)-colouring of F\mathcal{F} to be an rr-colouring of the sets without any monochromatic kk-chains F1F2FkF_1 \subset F_2 \subset \dots \subset F_k. We prove that for nn sufficiently large in terms of kk, the largest kk-chain-free families also maximise the number of (2,k)(2,k)-colourings. We also show that the middle level, ([n]n/2)\binom{[n]}{\lfloor n/2 \rfloor}, maximises the number of (3,2)(3,2)-colourings, and give asymptotic results on the maximum possible number of (r,k)(r,k)-colourings whenever r(k1)r(k-1) is divisible by three.Comment: 30 pages, final versio
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