A family of $r$ distinct sets $\{A_1,\ldots, A_r\}$ is an $r$-sunflower if for all $1 \leqslant i < j \leqslant r$ and $1 \leqslant i' < j' \leqslant r$, we have $A_i \cap A_j = A_{i'} \cap A_{j'}$. Erd\H{o}s and Rado conjectured in 1960 that every family $\mathcal{H}$ of $\ell$-element sets of size at least $K(r)^\ell$ contains an $r$-sunflower, where $K(r)$ is some function that depends only on $r$. We prove that if $\mathcal{H}$ is a family of $\ell$-element sets of VC-dimension at most $d$ and $|\mathcal{H}| > (C r (\log d+\log^\ast \ell))^\ell$ for some absolute constant $C > 0$, then $\mathcal{H}$ contains an $r$-sunflower. This improves a recent result of Fox, Pach, and Suk. When $d=1$, we obtain a sharp bound, namely that $|\mathcal{H}| > (r-1)^\ell$ is sufficient. Along the way, we establish a strengthening of the Kahn-Kalai conjecture for set families of bounded VC-dimension, which is of independent interest.

翻译：若一族$r$个互异集合$\{A_1,\ldots, A_r\}$满足对所有$1 \leqslant i < j \leqslant r$及$1 \leqslant i' < j' \leqslant r$均有$A_i \cap A_j = A_{i'} \cap A_{j'}$，则称其为一个$r$-向日葵。Erd\H{o}s与Rado于1960年猜想：每个包含至少$K(r)^\ell$个$\ell$元集合的族$\mathcal{H}$必然包含一个$r$-向日葵，其中$K(r)$是仅依赖于$r$的某个函数。本文证明：若$\mathcal{H}$是一个VC维至多为$d$的$\ell$元集合族，且对某个绝对常数$C > 0$有$|\mathcal{H}| > (C r (\log d+\log^\ast \ell))^\ell$，则$\mathcal{H}$包含一个$r$-向日葵。该结果改进了Fox、Pach与Suk近期的结论。当$d=1$时，我们得到了一个紧界：$|\mathcal{H}| > (r-1)^\ell$即为充分条件。在证明过程中，我们建立了有界VC维集合族情形下Kahn-Kalai猜想的强化形式，该结果具有独立的理论价值。