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} and Bell et al.~\cite{bcw} improved this bound to $(O(r \log(n))^n$. 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：以 $r$ 个花瓣构成的太阳花是指在一个基集 $X$ 上的一族 $r$ 个集合，使得 $X$ 中的每个元素要么不在任何集合中，要么在每一个集合中，要么恰好在一个集合中。Erdős 和 Rado \cite{er} 证明，若一个大小为 $n$ 的集族包含超过 $n!(r-1)^n$ 个集合，则该集族必含一个太阳花。Alweiss 等人 \cite{alwz} 以及后续的 Rao~\cite{rao} 和 Bell 等人 \cite{bcw} 将此上界改进为 $(O(r \log(n))^n$。本文研究集族中集合两两交集受限制的情形。特别地，当任意两个集合的交集大小属于集合 $L$ 时，我们改进了现有最佳上界。此外，利用 Alweiss 等人 \cite{alwz} 的技术，针对集合 $L$ 为不超过 $d$ 的非负整数这一特殊情况，我们给出了一个新上界。