Upper Bounds on the Multicolor Ramsey Numbers rk(C4)

The multicolor Ramsey number rk(C4) is the smallest integer N such that any k-edge coloring of KN contains a monochromatic C4. The current best upper bound of rk(C4) was obtained by Chung (1974) and independently by Irving (1974), i.e., rk(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 rk(C4) by showing that rk(C4) <= k2 + k - 1 for even k >= 6. The improvement is based on the upper bound of the Tur & aacute;n 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].