On Restricted Intersections and the Sunflower Problem

TL;DR

This paper investigates the structure of set families with restricted pairwise intersections, improving bounds on sunflower existence and providing new bounds for specific intersection constraints using advanced combinatorial techniques.

Contribution

It advances the understanding of sunflower bounds under intersection restrictions and introduces new bounds for cases with limited intersection sizes.

Findings

Improved bounds for set families with restricted pairwise intersections.

New bounds for the case where intersections are limited to integers up to d.

Enhanced techniques based on Alweiss et al. for specific intersection constraints.

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. Erd\H{o}s and Rado \cite{er} 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. \cite{alwz} and subsequently Rao~\cite{rao} and Bell et al.~\cite{bcw} 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. \cite{alwz}.