Diagonal Ramsey Numbers of Loose Cycles in Uniform Hypergraphs

A $k$-uniform loose cycle $\mathcal{C}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots, v_{(i-1)(k-1)+k}\}$, $1\leq i\leq n$, where we use mod $n(k-1)$ arithmetic. The diagonal Ramsey number of $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$, is asymptotically $\frac{1}{2}(2k-1)n$, as has been proved by Gyarfas, Sarkozy, and Szemeredi [Electron. J. Combin., 15 (2008), \#R126]. In this paper, we investigate to determine the exact value of $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ and we show that for $n\geq 2$ and $k\geq 8,$ $R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.$