A generalized Ramsey problem

Let f(n) be the minimum number such that there is a proper edge-coloring of Kn with f(n) colors with no path or cycle of 4 edges using one or two colors. It is shown that [(1+5)/2]n−3⩽f(n)⩽2n1+c/log n for a positive constants c. This improves the existent bounds on the variant of the Ramsey number f(n,5,9) studied by Erdős and Gyárfás.