tailieunhanh - Báo cáo toán học: "Set-Systems with Restricted Multiple Intersections"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Set-Systems with Restricted Multiple Intersections. | Set-Systems with Restricted Multiple Intersections Vince Grolmusz Department of Computer Science Eotvos University H-1117 Budapest HUNGARY E-mail grolmusz@ Submitted May 30 2001 Accepted February 20 2002. MR Subject Classifications 05D05 05C65 05D10 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 X k k V This result considerably strengthens an upper bound of Furedi 1983 and gives partial answer to a question of T. Sós 1976 . As an application we give a direct explicit construction for coloring the k-subsets of an n element set with t colors such that no monochromatic complete hypergraph on exp c log m 1 log log m 1 i-1 vertices exists. Keywords set-systems algorithmic constructions explicit Ramsey-graphs explicit Ramsey-hypergraphs 1 Introduction We are interested in set-systems with restricted intersection-sizes. The famous Ray-Chaudhuri-Wilson RCW75 and Frankl-Wilson FW81 theorems give strong upper bounds for the size of set-systems with restricted pairwise intersection sizes. T. Sos asked in 1976 Sos76 what happens if not the pairwise intersections but the k-wise intersection-sizes are restricted. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R8 1 Furedi Fur83 Ftir91 showed actually proving a much more general structure theorem that for d-uniform set-systems over an n element universe for very small d s d O loglog n the order of magnitude of the largest set-systems satisfying k-wise or just pairwise intersection restrictions are the same. In the present paper we strengthen this result of Furedi Fur83 . More exactly we prove the following k-wise version of the Deza-Frankl-Singhi theorem DFS83 . Note that no upper bounds for the sizes of sets in the .