For fixed integers p and q an edge coloring of K n is called a (p# q)-coloring if the edges of K n 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 K n . In  Erdos and Gy'arf'as studied this function, if p and q are fixed and n tends to infinity. They determined for every p the smallest q (= ; p 2 \Delta ; p + 3) for which f(n# p# q) is linear in n and the smallest q for which f(n# p# q)isquadraticinn. 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 weshow that that wecanhaveatmostlogp values of q which give a linear f(n# p# q). 1 Introduction 1.1 Notations and definitions For basic graph concepts see the monograph of Bollob'as . V (G)andE(G) denote the vertex-set and the edge-set of the graph G. ..