Sunflowers and Ramsey problems for restricted intersections

Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set $L\subseteq \{0,\dots,k-1\}$ and a family $\mathcal{F}$ of $k$-element sets which does not contain a sunflower with $m$ petals whose kernel size is in $L$, how large a subfamily of $\mathcal{F}$ can we find in which no pair has intersection size in $L$? We give matching upper and lower bounds, determining the dependence on $m$ for all $k$ and $L$. This problem also finds applications in quantum computing. As an application of our techniques, we also obtain a variant of F\"uredi's celebrated semilattice lemma, which is a key tool in the powerful delta-system method. We prove that one cannot remove the double-exponential dependency on the uniformity in F\"uredi's result, however, we provide an alternative with significantly better, single-exponential dependency on the parameters, which is still strong enough for most applications of the delta-system method.