Diagonal Ramsey numbers of loose cycles in uniform hypergraphs

TL;DR

This paper determines the exact diagonal Ramsey numbers for loose cycles in uniform hypergraphs, extending previous asymptotic results to precise values for certain parameters.

Contribution

It provides the exact value of the diagonal Ramsey number for loose cycles in hypergraphs for n ≥ 2 and k ≥ 8, advancing beyond prior asymptotic findings.

Findings

Exact formula for R(𝓒ₙ^k, 𝓒ₙ^k) when n ≥ 2 and k ≥ 8

Extension of asymptotic results to precise values

Improved understanding of hypergraph Ramsey numbers

Abstract

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 with 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 Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gy\'{a}rf\'{a}s, S\'{a}rk\"{o}zy and Szemer\'{e}di. In this paper, we investigate to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and we show that for $n\geq 2$ and $k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=(k-1)n+\lfloor\frac{n-1}{2}\rfloor.$$