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

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

\[
20 \leq R_2(C_4, C_4, C_4) \leq 21.
\]