Sunflowers and $L$-intersecting families

Abstract

Let $f(k,r,s)$ stand for the least number so that if $\mathcal{F}$ is an arbitrary $k$-uniform, $L$-intersecting set system, where $|L|=s$, and $\mathcal{F}$ has more than $f(k,r,s)$ elements, then $\mathcal{F}$ 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 $g(k,r,\ell )$.

10 pages