99 research outputs found

    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 log⁑2F(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)β‹…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

    The condensation transition in random hypergraph 2-coloring

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    For many random constraint satisfaction problems such as random satisfiability or random graph or hypergraph coloring, the best current estimates of the threshold for the existence of solutions are based on the first and the second moment method. However, in most cases these techniques do not yield matching upper and lower bounds. Sophisticated but non-rigorous arguments from statistical mechanics have ascribed this discrepancy to the existence of a phase transition called condensation that occurs shortly before the actual threshold for the existence of solutions and that affects the combinatorial nature of the problem (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). In this paper we prove for the first time that a condensation transition exists in a natural random CSP, namely in random hypergraph 2-coloring. Perhaps surprisingly, we find that the second moment method breaks down strictly \emph{before} the condensation transition. Our proof also yields slightly improved bounds on the threshold for random hypergraph 2-colorability. We expect that our techniques can be extended to other, related problems such as random k-SAT or random graph k-coloring
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