On the Size of Set Systems on [n] Not Containing Weak (r, Delta)-Systems

Letr?3 be an integer. A weak (r,?)-system is a family ofrsets such that all pairwise intersections among the members have the same cardinality. We show that fornlarge enough, there exists a family F of subsets of [n] such that F does not contain a weak (r,?)-system and |F|?2(1/3)·n1/5log4/5(r?1). This improves an earlier result of Erdo?s and Szemeredi (1978,J. Combin. Theory Ser. A24, 308?313; cf. Erdo?s, On some of my favorite theorems, in “Combinatorics, Paul Erdo?s Is Eighty,” Vol. 2, Bolyai Society Math. Studies, pp. 97?133, Janos Bolyai Math. Soc., Budapest, 1990).