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} improved this bound to $(O(r \log(rn))^n$. 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}.

翻译：带有$r$个花瓣的向日葵是基集$X$上$r个集合的族，使得$X$中每个元素要么不在任何集合中，要么在所有集合中，要么恰好在其中一个集合中。Erdős和Rado \cite{er}证明，若集合族中大小为$n$的集合数量超过$n!(r-1)^n$，则该族必含一个向日葵。Alweiss等人\cite{alwz}及后续Rao \cite{rao}将此界改进至$(O(r \log(rn))^n$。我们研究集合族中两两交集受限的情形。特别地，当任意两集合交集大小属于集合$L$时，我们改进了集合族的最佳已知界。此外，利用Alweiss等人\cite{alwz}的技术，针对集合$L$为不超过$d$的非负整数这一特例，我们给出了一个新的界。