Almost -rainbow edge-colorings of some small graphs

Let f(n, p, q) be the minimum number of colors necessary to color the edges of Kn so that every Kp is at least q-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdős and Gyárfás. We show that f(n, 5, 9) ≥ 7 4 n− 3, slightly improving the bound of Axenovich. We make small improvements on bounds of Erdős and Gyárfás by showing 5 6 n+1 ≤ f(n, 4, 5) and for all even n 6≡ 1(mod 3), f(n, 4, 5) ≤ n− 1. For a complete bipartite graphG = Kn,n, we show an n-color construction to color the edges of G so that every C4 ⊆ G is colored by at least three colors. This improves the best known upper bound of Axenovich, Füredi, and Mubayi.