Multiple vertex coverings by cliques

For positive integers m1,...,mk, let f(m1,...,mk) be the minimum order of a graph whose edges can be coloured with k colours such that every vertex is in a clique of cardinality mi all of whose edges have the i th colour for all i = 1,2,...,k. The value for k = 2 was determined by Entringer et al. (J. Graph Theory 24 (1997), 21–23). We show that if k is fixed then f(m,...,m) = 2km − o(m). We also provide some exact values for f(m,n,2)