82 research outputs found

    Multicolour Ramsey numbers of odd cycles

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    We show that for any positive integer rr there exists an integer kk and a kk-colouring of the edges of K2k+1K_{2^{k}+1} with no monochromatic odd cycle of length less than rr. This makes progress on a problem of Erd\H{o}s and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the kk-colour Ramsey number of the odd cycle and prove that, for all odd rr and all kk sufficiently large, there exists a constant ϵ=ϵ(r)>0\epsilon = \epsilon(r) > 0 such that Rk(Cr)>(r1)(2+ϵ)k1R_{k}(C_{r}) > (r-1)(2+\epsilon)^{k-1}

    An improvement on Łuczak's connected matchings method

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    A connected matching in a graph G is a matching contained in a connected component of G. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007)

    Improved bounds on the multicolor Ramsey numbers of paths and even cycles

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    We study the multicolor Ramsey numbers for paths and even cycles, Rk(Pn)R_k(P_n) and Rk(Cn)R_k(C_n), which are the smallest integers NN such that every coloring of the complete graph KNK_N has a monochromatic copy of PnP_n or CnC_n respectively. For a long time, Rk(Pn)R_k(P_n) has only been known to lie between (k1+o(1))n(k-1+o(1))n and (k+o(1))n(k + o(1))n. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of (k14+o(1))n(k - \frac{1}{4} + o(1))n. We improve the upper bound to (k12+o(1))n(k - \frac{1}{2}+ o(1))n. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest

    Ramsey properties of randomly perturbed graphs: cliques and cycles

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    Given graphs H1,H2H_1,H_2, a graph GG is (H1,H2)(H_1,H_2)-Ramsey if for every colouring of the edges of GG with red and blue, there is a red copy of H1H_1 or a blue copy of H2H_2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3,Kt)(K_3,K_t)-Ramsey (for t3t\ge 3). They also raised the question of generalising this result to pairs of graphs other than (K3,Kt)(K_3,K_t). We make significant progress on this question, giving a precise solution in the case when H1=KsH_1=K_s and H2=KtH_2=K_t where s,t5s,t \ge 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3,Kt)(K_3,K_t)-Ramsey question. Moreover, we give bounds for the corresponding (K4,Kt)(K_4,K_t)-Ramsey question; together with a construction of Powierski this resolves the (K4,K4)(K_4,K_4)-Ramsey problem. We also give a precise solution to the analogous question in the case when both H1=CsH_1=C_s and H2=CtH_2=C_t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs,Kt)(C_s,K_t)-Ramsey (for odd s5s\ge 5 and t4t\ge 4).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil Powierski, stated results for cliques in graphs of low positive density separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to appear in CP

    3‐Color bipartite Ramsey number of cycles and paths

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    The k-colour bipartite Ramsey number of a bipartite graph H is the least integer n for which every k-edge-coloured complete bipartite graph Kn,n contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gy´arf´as and Lehel, who determined the 2-colour Ramsey number of paths. In this paper we determine asymptotically the 3-colour bipartite Ramsey number of paths and (even) cycles
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