The Ramsey Number of Diamond-Matchings and Loose Cycles in Hypergraphs

The 2-color Ramsey number R(C3 n, C3 n) of a 3-uniform loose cycle Cn is asymptotic to 5n/4 as has been recently proved by Haxell, Luczak, Peng, Rödl, Ruciński, Simonovits and Skokan. Here we extend their result to the r-uniform case by showing that the corresponding Ramsey number is asymptotic to (2r−1)n 2r−2 . Partly as a tool, partly as a subject of its own, we also prove that for r ≥ 2, R(kDr, kDr) = k(2r − 1)− 1 and R(kDr, kDr, kDr) = 2kr − 2 where kDr is the hypergraph having k disjoint copies of two r-element hyperedges intersecting in two vertices. Research supported in part by OTKA Grant No. K68322. Research supported in part by the National Science Foundation under Grant No. DMS-0456401, by OTKA Grant No. K68322 and by a Janos Bolyai Research Scholarship. the electronic journal of combinatorics 15 (2008), #R126 1