Diagonal ramsey numbers of loose cycles in uniform hypergraphs

Abstract

A κ-uniform loose cycle Ckn is a hypergraph with vertex set v1, v2, ... , vn(κ-1) and the set of n edges ei = v(i-1)(κ-1)+1; v(i-1)(κ-1)+2; ... ; v(i-1)(κ-1)+kg, 1 ≤ i ≤ n, where we use mod n(κ - 1) arithmetic. The diagonal Ramsey number of Ckn , R(Ckn ; Ckn ), is asymptotically 1/2 (2κ-1)n, as has been proved by Gyárfás, Sárközy, and Szemerédi [Electron. J. Combin., 15 (2008), #R126]. In this paper, we investigate to determine the exact value of R(Ckn , Ckn ) and we show that for n ≥ 2 and k ≥ 8, R(Ckn , Ckn ) = (κ - 1)n + ⌊ n-1/2 ⌋. © 2017 Society for Industrial and Applied Mathematics.