Upper Bounds on the Multicolor Ramsey Numbers rk(C4)

The multicolor Ramsey number r_k(C₄) is the smallest integer N such that any k-edge coloring of K_N contains a monochromatic C₄. The current best upper bound of r_k(C₄) was obtained by Chung (1974) and independently by Irving (1974), i.e., r_k(C₄) ≤ k² + k + 1 for all k ≥ 2. There is no progress on the upper bound since then. In this paper, we improve the upper bound of r_k(C₄) by showing that r_k(C₄) ≤ k² + k − 1 for even k ≥ 6. The improvement is based on the upper bound of the Turán number ex(n, C₄), in which we mainly use the double counting method and many novel ideas from Firke, Kosek, Nash, and Williford [J. Combin. Theory, Ser. B 103 (2013), 327–336].