Abstract
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 Erdýos and Gyarfas. We show that f(n,5,9) ≥ 7 n−3, slightly improving the bound of Axenovich. We make small improvements on bounds of Erdýos and Gyarfas by showing 5 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 graph G = 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, Furedi, and Mubayi.
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