Ramsey numbers of $5$-uniform loose cycles

TL;DR

This paper investigates the Ramsey numbers of 5-uniform loose cycles, confirming a conjecture for large cycle lengths and expanding understanding of hypergraph Ramsey theory.

Contribution

It proves the conjecture for 5-uniform loose cycles when the cycle length is sufficiently large, advancing the knowledge of hypergraph Ramsey numbers.

Findings

Confirmed the conjecture for k=5 and large n

Extended the understanding of Ramsey numbers for loose cycles

Provided exact values for specific hypergraph configurations

Abstract

Gy\'{a}rf\'{a}s et al. determined the asymptotic value of the diagonal Ramsey number of $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n),$ generating the same result for $k=3$ due to Haxell et al. Recently, the exact values of the Ramsey numbers of 3-uniform loose paths and cycles are completely determined. These results are motivations to conjecture that for every $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_m)=(k-1)n+\lfloor\frac{m-1}{2}\rfloor,$$ as mentioned by Omidi et al. More recently, it is shown that this conjecture is true for $n=m\geq 2$ and $k\geq 7$ and for $k=4$ when $n>m$ or $n=m$ is odd. Here we investigate this conjecture for $k=5$ and demonstrate that it holds for $k=5$ and sufficiently large $n$.