Ramsey Numbers of 5-Uniform Loose Cycles

Abstract

Gyárfás 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 $n\ge m\ge 3$ and $k\ge 3,$

$$\begin{aligned} R(\mathcal {C}^k_n,\mathcal {C}^k_m)=(k-1)n+\Big \lfloor \frac{m-1}{2}\Big \rfloor , \end{aligned}$$

as mentioned by Omidi et al. More recently, it has been shown that this conjecture is true for $n=m\ge 2$ and $k\ge 7$ 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.