UpperBoundsontheMulticolorRamseyNumbersrk（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 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].