Split and balanced colorings of complete graphs

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K, in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph Ks has a coloring which is not (r,n)-split is denoted by f,.(n). Balanced (r,n)-colorings are defined as edge r-colorings of Ks such that every subset of IN/r] vertices contains a monochromatic K~ in all colors. Then yr(n) is defined as the smallest N such that Ks has a balanced (r, n)-coloring. The definitions imply that fi(n)<~gr(n). The paper gives estimates and exact values of these functions for various choices of parameters. (~) 1999 Elsevier Science B.V. All rights reserved