Title: Restricted Intersections and the Sunflower Problem
| Speaker: | Jeremy Chizewer |
| Affiliation: | University of Waterloo |
| Location: | MC 5479 |
Abstract: 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. Erdos and Rado 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. and subsequently Rao improved this bound to $(O(r \log(rn))^n$.
I will talk about the case where the pairwise intersections of the set family are restricted. In particular, I will talk about an improved bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. I will 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.