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

Abstract

The $2$-color Ramsey number $R({\cal{C}}_n^3,{\cal{C}}_n^3)$ of a $3$-uniform loose cycle ${\cal{C}}_n$ 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 ${(2r-1)n\over 2r-2}$. Partly as a tool, partly as a subject of its own, we also prove that for $r\ge 2$, $R(kD_r,kD_r)=k(2r-1)-1$ 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.