Suppose the r-subsets of an n-element set are colored by t colors. THEOREM 1.1. IJ n ~ (t l)(k 1) + k . r, then there are k pairwise disjoint r-sets having the same color. This was conjectured by Erdos [El in 1973. Let T(n, r, s) denote the 'lUran number for s-uniform hypergraphs (see §1). THEOREM 1.3. IJe > 0, t ~ (1-e)T(n,r,s)/(k-1), andn > no(e,r,s,k), then there are k r-sets At, A2, .. . , Ak having the same color such that IAi n Ajl < s Jor all 1 ~ i < j ~ k. IJ s = 2, e can be omitted… CONTINUE READING