发表机构
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University; Frontiers Science Center for Nonlinear Expectations, Ministry of Education（山东大学数学交叉科学研究中心; 教育部非线性期望前沿科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有界VC维集合系统中阈值受期望阈值乘以对数因子控制，验证Talagrand猜想，并改进向日葵界。
AI 中文摘要
设$p_c(\mathcal F)$、$q(\mathcal F)$和$q_f(\mathcal F)$分别表示有限集非空子集族$\mathcal F$的阈值、期望阈值和分数期望阈值。我们证明存在绝对常数$C>0$，使得若$\mathcal F$的VC维至多为$d\ge1$，则$p_c(\mathcal F)\le Cq(\mathcal F)\log(d+1)$。更一般地，对于每个$0<\varepsilon\le1/2$，密度为$\min\{1,Cq(\mathcal F)\log((d+1)/\varepsilon)\}$的二项随机集合以至少$1-\varepsilon$的概率包含$\mathcal F$的一个成员。因此，$q_f(\mathcal F)\le Cq(\mathcal F)\log(d+1)$，验证了Talagrand的积分-分数猜想对任意固定VC维的族成立。我们还证明，若一个$k$-扩散概率测度的支撑集的VC维至多为$d$，则密度为$\min\{1,(C/k)\log((d+1)/\varepsilon)\}$的二项随机集合以至少$1-\varepsilon$的概率包含其支撑集的一个成员。在这两个随机包含结果中，因子$\log((d+1)/\varepsilon)$在绝对常数意义下是最优的。作为扩散定理的应用，我们证明每个VC维至多为$d$且成员数多于$(C p^{-1}\log((d+1)/\varepsilon))^n$的$n$-一致族包含一个$(p,\varepsilon)$-鲁棒向日葵。特别地，每个成员数多于$(Cr\log(d+1))^n$的此类族包含一个$r$-向日葵，改进了Ge、Wang、Xu和Zhao最近的界$(Crd)^n$。
英文摘要
Let $p_c(\mathcal F)$, $q(\mathcal F)$, and $q_f(\mathcal F)$ denote the threshold, expectation threshold, and fractional expectation threshold of a family $\mathcal F$ of nonempty subsets of a finite set, respectively. We prove that there is an absolute constant $C>0$ such that, if $\mathcal F$ has VC dimension at most $d\ge1$, then $p_c(\mathcal F)\le Cq(\mathcal F)\log(d+1)$. More generally, for every $0<\varepsilon\le1/2$, a binomial random set of density $\min\{1,Cq(\mathcal F)\log((d+1)/\varepsilon)\}$ contains a member of $\mathcal F$ with probability at least $1-\varepsilon$. Consequently, $q_f(\mathcal F)\le Cq(\mathcal F)\log(d+1)$, verifying Talagrand's integral--fractional conjecture for families of any fixed VC dimension. We also prove that if a $k$-spread probability measure has support of VC dimension at most $d$, then a binomial random set of density $\min\{1,(C/k)\log((d+1)/\varepsilon)\}$ contains a member of its support with probability at least $1-\varepsilon$. In both random-containment results, the factor \(\log((d+1)/\varepsilon)\) is optimal up to absolute constants. As an application of the spread theorem, we prove that every $n$-uniform family of VC dimension at most $d$ with more than $(C p^{-1}\log((d+1)/\varepsilon))^n$ members contains a $(p,\varepsilon)$-robust sunflower. In particular, every such family with more than $(Cr\log(d+1))^n$ members contains an $r$-sunflower, improving the recent bound $(Crd)^n$ of Ge, Wang, Xu, and Zhao.
Comments20 pages, Happy Mid-Autumn Festival, zhong qiu kuai le
