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Constructive Ramsey Numbers for Loose Hyperpaths

  • Andrzej Dudek
  • Andrzej Ruciński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

For positive integers k and \(\ell \), a k-uniform hypergraph is called a loose path of length \(\ell \), and denoted by \(P_\ell ^{(k)}\), if its vertex set is \(\{v_1, v_2, \ldots , v_{(k-1)\ell +1}\}\) and the edge set is \(\{e_i = \{ v_{(i-1)(k-1)+q} : 1 \le q \le k \},\ i=1,\dots ,\ell \}\), that is, each pair of consecutive edges intersects on a single vertex. Let \(R(P_\ell ^{(k)};r)\) be the multicolor Ramsey number of a loose path that is the minimum n such that every r-edge-coloring of the complete k-uniform hypergraph \(K_n^{(k)}\) yields a monochromatic copy of \(P_\ell ^{(k)}\). In this note we are interested in constructive upper bounds on \(R(P_\ell ^{(k)};r)\) which means that on the cost of possibly enlarging the order of the complete hypergraph, we would like to efficiently find a monochromatic copy of \(P_\ell ^{(k)}\) in every coloring. In particular, we show that there is a constant \(c>0\) such that for all \(k\ge 2\), \(\ell \ge 3\), \(2\le r\le k-1\), and \(n\ge k(\ell +1)r(1+\ln (r))\), there is an algorithm such that for every r-edge-coloring of the edges of \(K_n^{(k)}\), it finds a monochromatic copy of \(P_\ell ^{(k)}\) in time at most \(cn^k\).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsWestern Michigan UniversityKalamazooUSA
  2. 2.Department of Discrete MathematicsAdam Mickiewicz UniversityPoznańPoland

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