Intersection Statements for Systems of Sets

A family ofrsets is called aΔ-system if any two sets have the same intersection. Denote byF(n, r) the most number of subsets of ann-element set which do not contain aΔ-system consisting ofrsets. Constructive new lower bounds forF(n, r) are given which improve known probabilistic results, and a new upper bound is given by employing an argument due to Erdős and Szemerédi. Another construction is given which shows that for certainn,F(n, 3)⩾1.551n−2. We also show a relationship between the upper bound forF(n, 3) and the Erdős–Rado conjecture on the largest uniform family of sets not containing aΔ-system.