Sunflowers and $L$-intersecting families

TL;DR

This paper establishes upper bounds on the size of certain set systems that guarantee the existence of sunflowers with a specified number of petals, advancing combinatorial understanding of intersecting families.

Contribution

It provides new upper bounds for functions related to sunflower existence in $L$-intersecting and $ ext{ell}$-intersecting set systems, generalizing previous results.

Findings

Derived an upper bound for $f(k,3,s)$.

Established an upper bound for $g(k,r, ext{ell})$.

Extended sunflower existence conditions to broader intersecting set systems.

Abstract

Let $f(k,r,s)$ stand for the least number so that if $\cal F$ is an arbitrary $k$-uniform, $L$-intersecting set system, where $|L|=s$, and $\cal F$ has more than $f(k,r,s)$ elements, then $\cal F$ contains a sunflower with $r$ petals. We give an upper bound for $f(k,3,s)$. Let $g(k,r,\ell)$ be the least number so that any $k$-uniform, $\ell$-intersecting set system of more than $g(k,r,\ell)$ sets contains a sunflower with $r$ petals. We give also an upper bound for $g(k,r,\ell)$.