Abstract

We give a generalization for the Deza-Frankl-Singhi Theorem in case of multiple intersections. More exactly, we prove, that if H is a set-system, which satisfies that for some k, the k-wise intersections occupy only ` residue-classes modulo a p prime, while the sizes of the members of H are not in these residue classes, then the size of H is at most (k \Gamma 1) ` X i=0 / n i ! This result considerably strengthens an upper bound of Furedi (1983), and gives partial answer to a question of T. S'os (1976). As an application, we give explicit constructions for coloring the k-subsets of an n element set with t colors, such that no monochromatic complete hypergraph on exp(c k (log n log log n) 1=t ) vertices exists. By our best knowledge, this is the first explicit construction of a Ramsey-hypergraph.

Citations