A note on multicolor Ramsey number of small odd cycles versus a large clique

TL;DR

This paper establishes improved lower bounds for multicolor Ramsey numbers involving small odd cycles versus large cliques, using probabilistic constructions, advancing previous bounds in combinatorics.

Contribution

It provides new lower bounds for $R_k(C_5;K_m)$ and $R_k(C_7;K_m)$, refining earlier estimates through probabilistic methods.

Findings

Lower bounds for $R_k(C_5;K_m)$ and $R_k(C_7;K_m)$ are established.

Bounds are expressed as functions of $m$, $k$, and logarithmic factors.

Results improve upon previous bounds by Alon and R"{o}dl.

Abstract

Let $R_k(H;K_m)$ be the smallest number $N$ such that every coloring of the edges of $K_{N}$ with $k+1$ colors has either a monochromatic $H$ in color $i$ for some $1\leqslant i\leqslant k$, or a monochromatic $K_{m}$ in color $k+1$. In this short note, we study the lower bound for $R_k(H;K_m)$ when $H$ is $C_5$ or $C_7$, respectively. We show that \begin{equation*} R_{k}(C_5;K_m)=\Omega(m^{\frac{3k}{8}+1}/(\log{m})^{\frac{3k}{8}+1}), \end{equation*} and \begin{equation*} R_{k}(C_7;K_m)=\Omega(m^{\frac{2k}{9}+1}/(\log{m})^{\frac{2k}{9}+1}), \end{equation*} for fixed positive integer $k$ and $m\rightarrow\infty$. These slightly improve the previously known lower bound $R_{k}(C_{2\ell+1};K_m)=\Omega(m^{\frac{k}{2\ell-1}+1}/(\log m)^{k+\frac{2k}{2\ell-1}})$ obtained by Alon and R\"{o}dl. The proof is based on random block constructions and random blowups argument.