On Restricted Intersections and the Sunflower Problem
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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 <cit.> 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. <cit.> and subsequently Rao <cit.> and Bell et al. <cit.> 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. <cit.>.
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