On Restricted Intersections and the Sunflower Problem

A sunflower with r petals is a collection of r sets over a ground set X such that every element in X is in no set, every set, or exactly one set. Erdős and Rado [5] showed that a family of sets of size n contains a sunflower if there are more than \(n!(r-1)^n\) sets in the family. Alweiss et al. [1] and subsequently, Rao [7] and Bell et al. [2] improved this bound to \((O(r \log n))^n\). We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best known bound for set families when the size of the pairwise intersections of any two sets is in a set L. We also present a new bound for the special case when the set L is the nonnegative integers less than or equal to d using the techniques of Alweiss et al. [1].