Sunflowers and $L$-intersecting families

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 $g(k,r,\ell)$.