1,469 research outputs found

    Monochromatic loose paths in multicolored kk-uniform cliques

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    For integers k2k\ge 2 and 0\ell\ge 0, a kk-uniform hypergraph is called a loose path of length \ell, and denoted by P(k)P_\ell^{(k)}, if it consists of \ell edges e1,,ee_1,\dots,e_\ell such that eiej=1|e_i\cap e_j|=1 if ij=1|i-j|=1 and eiej=e_i\cap e_j=\emptyset if ij2|i-j|\ge2. In other words, each pair of consecutive edges intersects on a single vertex, while all other pairs are disjoint. Let R(P(k);r)R(P_\ell^{(k)};r) be the minimum integer nn such that every rr-edge-coloring of the complete kk-uniform hypergraph Kn(k)K_n^{(k)} yields a monochromatic copy of P(k)P_\ell^{(k)}. In this paper we are mostly interested in constructive upper bounds on R(P(k);r)R(P_\ell^{(k)};r), meaning that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of P(k)P_\ell^{(k)} in every coloring. In particular, we show that there is a constant c>0c>0 such that for all k2k\ge 2, 3\ell\ge3, 2rk12\le r\le k-1, and nk(+1)r(1+ln(r))n\ge k(\ell+1)r(1+\ln(r)), there is an algorithm such that for every rr-edge-coloring of the edges of Kn(k)K_n^{(k)}, it finds a monochromatic copy of P(k)P_\ell^{(k)} in time at most cnkcn^k. We also prove a non-constructive upper bound R(P(k);r)(k1)rR(P_\ell^{(k)};r)\le(k-1)\ell r

    On the minimum degree of minimal Ramsey graphs for multiple colours

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    A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s_2 was initiated by Burr, Erd\H{o}s, and Lov\'{a}sz in 1976 when they showed that for the clique s_2(K_k)=(k-1)^2. In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k) = r^2 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r

    Combinatorial theorems relative to a random set

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    We describe recent advances in the study of random analogues of combinatorial theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201

    Density version of the Ramsey problem and the directed Ramsey problem

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    We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on nn vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges ERB|E_{RB}| is given. The aim is to find the maximal size ff of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on nn vertices such that the number of bi-oriented edges Ebi|E_{bi}| is given. The aim is to bound the size FF of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if ERB=Ebi=p(n2)|E_{RB}|=|E_{bi}| = p{n\choose 2}, (p[0,1)p\in [0,1)), then f,F<Cplog(n)f, F < C_p\log(n) where CpC_p only depend on pp, while if m=(n2)ERB<n3/2m={n \choose 2} - |E_{RB}| <n^{3/2} then f=Θ(n2m+n)f= \Theta (\frac{n^2}{m+n}). The latter case is strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| = p{n\choose 2}

    Ramsey numbers of ordered graphs

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    An ordered graph is a pair G=(G,)\mathcal{G}=(G,\prec) where GG is a graph and \prec is a total ordering of its vertices. The ordered Ramsey number R(G)\overline{R}(\mathcal{G}) is the minimum number NN such that every ordered complete graph with NN vertices and with edges colored by two colors contains a monochromatic copy of G\mathcal{G}. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings Mn\mathcal{M}_n on nn vertices for which R(Mn)\overline{R}(\mathcal{M}_n) is superpolynomial in nn. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number R(G)\overline{R}(\mathcal{G}) is polynomial in the number of vertices of G\mathcal{G} if the bandwidth of G\mathcal{G} is constant or if G\mathcal{G} is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric
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