Intersection Statements for Systems of Sets

A family ofrsets is called a?-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 a?-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 Erdo?s and Szemeredi. 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 Erdo?s?Rado conjecture on the largest uniform family of sets not containing a?-system.

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