有限偏序集的间隙
On the Gap of Finite Posets
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对有限偏序集，证明了宽度与期望秩间隙的猜想关系，构造了宽度2偏序集使间隙任意大，还构建了满足熵与间隙特定关系的偏序集，核心思想来自ChatGPT 5.6 Sol。
AI中文摘要：
设P是一个具有n个元素的有限非空偏序集，f:P→{1,…,n}是一个均匀随机的保序双射，令h_P(x)=E[f(x)]。定义gap(P)为包含0、n+1以及所有期望秩h_P(x)的有序列表中相邻值之间的最大差值。记w(P)为两两不可比子集的最大大小。我们证明了三个结果：其一，证明了Brightwell与Trotter（2002）、Biró与Trotter（2011）以及Aires与Kahn（2025）的研究中反复出现且形式逐渐一般化的关于宽度与期望秩间隙的古老猜想关系：gap(P)≤2w(P)-1；其二，对任意L>0，构造了一个宽度为2的偏序集，使得其每条极大链的期望秩间隙至少为L，计算该间隙时包含两个端点间距；其三，对任意r∈N，构造了偏序集P_r，使得每个非空选定集合X诱导的相对序具有低于3|X|的以2为底的熵，而gap(P_r)≥(3/2)^r，因此间隙可以任意大，同时每个选定集合上的诱导序具有相对较小的熵。所有三个结果的核心思想均由ChatGPT 5.6 Sol发现。
英文摘要：
Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
