The Ramsey number of diamond-matchings and loose cycles in hypergraphs
Summary: The 2-color Ramsey number \(R({\mathcal{C}}_n^3,{\mathcal{C}}_n^3)\) of a 3-uniform loose cycle \({\mathcal{C}}_n\) is asymptotic to \(5n/4\) as has been recently proved by Haxell, Łuczak, 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 \(\frac{(2r-1)n}{2r-2}\). Partly as a tool, partly as a subject of its own, we also prove that for \(r\geq 2, R(kD_r,kD_r)=k(2r-1)-1\) and \(R(kD_r,kD_r,kD_r)=2kr-2\) where \(kD_r\) is the hypergraph having \(k\) disjoint copies of two \(r\)-element hyperedges intersecting in two vertices.
scientific article; zbMATH DE number 5541031