AI 中文总结
本文证明了任意非平凡递增族上期望阈值与分数期望阈值之间的对数星比较，给出一个以对数星函数为界的上界不等式。
AI 中文摘要
我们证明了对于有限基础集合 $V$（大小为 $|V|=N$）上的任意非平凡递增族 $\mathcal F$，期望阈值 $q(\mathcal F)$ 与分数期望阈值 $q_f(\mathcal F)$ 之间存在一个对数星比较。具体而言，我们证明了 $q_f(\mathcal F)\le64\log_2^*(N+2)\\,q(\mathcal F)$，其中 $\log_2^* x$ 定义为使得对 $x$ 反复应用 $\log_2$ 共 $k$ 次后所得结果不超过 1 的最小非负整数 $k$。
英文摘要
We proved a log-star comparison between the expectation threshold $q(\mathcal F)$ and the fractional expectation threshold $q_f(\mathcal F)$ for any nontrivial increasing family $\mathcal F$ on a finite ground set $V$ of size $|V|=N$. Specifically, we show that $$q_f(\mathcal F)\le64\log_2^*(N+2)\,q(\mathcal F),$$ where $\log_2^* x$ is the iterated logarithm of base-2. Using similiar methods, one can extend this result to the following stronger form: there exists a universal constant $C>0$ such that \[ q_f(\mathcal F)\le Cq(\mathcal F)\log_2^*(l(\mathcal F)),\quad q_f(\mathcal F)\le Cq(\mathcal F)\log_2^*(\operatorname{VC}(\min\mathcal F)), \] where $l(\mathcal F):=\max\{2,\max_{S\in\min\mathcal F}|S|\}$ and $\operatorname{VC}(\min\mathcal F)$ denotes the VC-dimension of $\min\mathcal F$. On October 7, 2026, OpenAI released a result proving the equivalence of fractional and integral thresholds, which supersedes the findings in this note. The authors are keeping this note for historical reference.
