Edge-colored complete graphs with precisely colored subgraphs

Letf(s, t; k) be the largest value ofm such that it is possible tok-color the edges of the complete graphK _m so that everyK _s ⊆K _m has exactlyt colors occuring on its edges. The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large. Dedicated to Paul Erdős on his seventieth birthday