ON THE SIZE OF SET SYSTEMS ON [N] NOT CONTAINING WEAK (R, DELTA)-SYSTEMS

Let r greater than or equal to 3 be an integer. A weak (r, Delta)-syst em is a family of r sets such that all pairwise intersections among th e members have the same cardinality. We show that for n large enough. there exists a family F of subsets of [n] such that F does not contain a weak (r, Delta)-system and \F\ greater than or equal to 2((1/3).n1/ 5.log4/5(r-1)). This improves an earlier result of Erdos and Szemeridi (1978, J. Combin. Theory Ser. A 24, 308-313: cf. Erdos. On some of my favorite theorems, in ''Combinatories, Paul Erdos Is Eighty,'' Vol. 2 , Bolyai Society Math. Studies, pp. 97-133, Janos Bolyai Math. Soc. Bu dapest. 1990). (C) 1997 Academic Press.