On Edge Colorings with at Least q Colors in Every Subset of p Vertices

For fixed integers p and q, an edge coloring of Kn is called a (p, q)-coloring if the edges of Kn in every subset of p vertices are colored with at least q distinct colors. Let f(n, p, q) be the smallest number of colors needed for a (p, q)-coloring of Kn. In [3] Erdős and Gyárfás studied this function if p and q are fixed and n tends to infinity. They determined for every p the smallest q (= (p 2 ) − p + 3) for which f(n, p, q) is linear in n and the smallest q for which f(n, p, q) is quadratic in n. They raised the question whether perhaps this is the only q value which results in a linear f(n, p, q). In this paper we study the behavior of f(n, p, q) between the linear and quadratic orders of magnitude. In particular we show that that we can have at most log p values of q which give a linear f(n, p, q).