AI 中文总结
本文证明了所有至多4个点的偏序集的饱和数要么有界要么线性，并特别解决了由V形加孤立点构成的偏序集的线性饱和问题，揭示了其饱和族的结构。
AI 中文摘要
给定一个有限偏序集 $\mathcal P$，其诱导饱和数 $\text{sat}^*(n,\mathcal P)$ 是 $[n]$ 的子集族所能具有的最小大小，使得该子集族不包含 $\mathcal P$ 的诱导副本，但向其中添加任何其他集合都会产生一个诱导副本。饱和数已经显示出一种尖锐的二分法——Freschi、Piga、Sharifzadeh 和 Treglown 表明，对于任何给定的偏序集，其饱和数要么有界，要么至少为 $2\sqrt n$。主导猜想是，事实上，饱和数要么有界，要么恰好是线性的。在本文中，我们证明了这一二分法对所有至多4个点的偏序集成立。这些偏序集中的大多数，尤其是蝴蝶形和菱形，此前各自长期难以分析，而本文所处理的缺失部分是由一个 $\mathcal V$ 和一个孤立点组成的偏序集，记为 $\widehat{\mathcal V}$，我们证明了其饱和数是线性的。证明的架构揭示了任意 $\widehat{\mathcal V}$-饱和族的结构，即一个反链，其上方至多有两条链，下方是一个倒置的森林。此外，我们预期，至少部分地，这一结构与更广泛的包含孤立点的偏序集类别相关。
英文摘要
Given a finite poset $\mathcal P$, its induced saturation number, $\text{sat}^*(n,\mathcal P)$ is the smallest size a family of subsets of $[n]$ can have such that it does not contain an induced copy of $\mathcal P$, but adding any other set to it creates one. The saturation numbers already show a sharp dichotomy -- Freschi, Piga, Sharifzadeh and Treglown showed that for any given poset its saturation number is either bounded or at least $2\sqrt n$. The dominant conjecture is that in fact, the saturation numbers are either bounded, or exactly linear. In this paper we show that this dichotomy is true for all posets on at most 4 points. Most of these posets, most notably the butterfly and the diamond, had individually long resisted analysis, and the missing piece which this paper tackles is the poset comprised of a $\mathcal V$ and an isolated point, denoted by $\widehat{\mathcal V}$, for which we show linear saturation. The architecture of the proof uncovers the structure of an arbitrary $\widehat{\mathcal V}$- saturated family, namely an antichain with at most two chains above it, and an upside-down forest below it. Moreover, we anticipate that, at least partially, this structure is relevant to the wider class of posets that contain an isolated point.
Comments15 pages, 19 figures
