7,260 research outputs found

    Erdos-Szekeres-type theorems for monotone paths and convex bodies

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    For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples (j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a monotone path of length n. Given any integers n\ge k\ge 2 and q\ge 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]=\{1,2,..., N\} with q colors, we can always find a monochromatic monotone path of length n? Denoting this minimum by N_k(q,n), it follows from the seminal 1935 paper of Erd\H os and Szekeres that N_2(q,n)=(n-1)^q+1 and N_3(2,n) = {2n -4\choose n-2} + 1. Determining the other values of these functions appears to be a difficult task. Here we show that 2^{(n/q)^{q-1}} \leq N_3(q,n) \leq 2^{n^{q-1}\log n}, for q \geq 2 and n \geq q+2. Using a stepping-up approach that goes back to Erdos and Hajnal, we prove analogous bounds on N_k(q,n) for larger values of k, which are towers of height k-1 in n^{q-1}. As a geometric application, we prove the following extension of the Happy Ending Theorem. Every family of at least M(n)=2^{n^2 \log n} plane convex bodies in general position, any pair of which share at most two boundary points, has n members in convex position, that is, it has n members such that each of them contributes a point to the boundary of the convex hull of their union.Comment: 32 page

    A combinatorial proof of the extension property for partial isometries

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    We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes Mathematicae Universitatis Carolina

    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

    The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

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    Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl

    Monochromatic connected matchings in 2-edge-colored multipartite graphs

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    A matching MM in a graph GG is connected if all the edges of MM are in the same component of GG. Following \L uczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 22-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.Comment: 29 pages, 2 figure
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