Intersections ofk-element sets

Abstract

LetF be a collection ofk-element sets with the property that the intersection of no two should be included in a third. We show that such a collection of maximum size satisfies .2715k+o(k)≦≦log₂ |F|≦.7549k+o(k) settling a question raised by Erdős. The lower bound is probabilistic, the upper bound is deduced via an entropy argument. Some open questions are posed.