INTERSECTION STATEMENTS FOR SYSTEMS OF SETS

A Family of r sets is called a Delta-system if any two sets have the s ame intersection. Denote by F(n, r) the most number of subsets of an n -element set which do not contain a Delta-system consisting of r sets. Constructive new lower bounds for F(n,r) are given which improve know n probabilistic results, and a new upper bound is given by employing a n argument due to Erdos and Szemeredi. Another construction is given w hich shows that for certain n, F(n, 3)greater than or equal to 1.551(n -2). We also show a relationship between the upper bound for F(n, 3) a nd the Erdos-Rado conjecture on the largest uniform family of sets not containing a Delta-system. (C) 1997 Academic Press.