Algebraic and Enumerative Combinatorics Seminar - Jeremy Chizewer

Title: The Sunflower Problem: Restricted Intersections

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

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 and Bell et al. improved this bound to $(O(r \log(n))^n$.

In this talk, I will discuss the sunflower problem with an additional restriction, a bound on the size of pairwise intersections in the set family. In particular, I will show an improved bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. This talk is based on https://arxiv.org/abs/2307.01374.