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

Letr⩾3 be an integer. A weak (r, Δ)-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, Δ)-system and |F|⩾2(1/3)·n1/5 log4/5(r−1). This improves an earlier result of Erdős and Szemerédi (1978,J. Combin. Theory Ser. A24, 308–313; cf. Erdős, On some of my favorite theorems, in “Combinatorics, Paul Erdős Is Eighty,” Vol. 2, Bolyai Society Math. Studies, pp. 97–133, János Bolyai Math. Soc., Budapest, 1990).