Diagonal Ramsey numbers of loose cycles in uniform hypergraphs

Abstract

A $k$-uniform loose cycle ${\mathcal{C}}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\dots ,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\dots ,v_{(i-1)(k-1)+k}\}$, $1\le i\le n$, where we use mod $n(k-1)$ arithmetic. The Ramsey number $R({\mathcal{C}}_n^k,{\mathcal{C}}_n^k)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gyárfás, Sárközy and Szemerédi. In this paper, we investigate to determining the exact value of diagonal Ramsey number of ${\mathcal{C}}_n^k$ and we show that for $n\ge 2$ and $k\ge 8$ $$R({\mathcal{C}}_n^k,{\mathcal{C}}_n^k)=(k-1)n+\lfloor \frac{n-1}{2}\rfloor .$$