A Note on the 2-Ramsey Numbers of 4-Cycles

Abstract

A balanced complete bipartite graph is a complete bipartite graph where the degrees of its vertices differ by at most 1. In a red-blue-green coloring of the edges of a graph $G$, every edge of $G$ is colored red, blue, or green. For three graphs $F_1$, $F_2$, and $F_3$, the 2-Ramsey number $R_2(F_1,F_2,F_3)$ of $F_1$, $F_2$, and $F_3$, if it exists, is the smallest order of a balanced complete bipartite graph $G$ such that every red-blue-green coloring of the edges of $G$ contains a red $F_1$, a blue $F_2$, or a green $F_3$. In this note, we determine that

$$20\le R_2(C_4,C_4,C_4)\le 21.$$