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    Hypergraph Ramsey numbers

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    The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq 2^{n^{c \log n}}. Finally, we make some progress on related hypergraph Ramsey-type problems

    On globally sparse Ramsey graphs

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    We say that a graph GG has the Ramsey property w.r.t.\ some graph FF and some integer r2r\geq 2, or GG is (F,r)(F,r)-Ramsey for short, if any rr-coloring of the edges of GG contains a monochromatic copy of FF. R{\"o}dl and Ruci{\'n}ski asked how globally sparse (F,r)(F,r)-Ramsey graphs GG can possibly be, where the density of GG is measured by the subgraph HGH\subseteq G with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs FF. In this work we determine the Ramsey density up to some small error terms for several cases when FF is a complete bipartite graph, a cycle or a path, and r2r\geq 2 colors are available

    New Computational Upper Bounds for Ramsey Numbers R(3,k)

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    Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds. Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3,k) are obtained by completing the computation of the exact values of e(3,k,n) for all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The enumeration of all graphs witnessing the values of e(3,k,n) is completed for all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35 vertices is unique up to isomorphism. For the case of R(3,10), first we establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently, that if R(3,10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for R(3,11); added some note
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