循环 Poset 中的禁止子集

Order Pub Date : 2024-08-26 DOI:10.1007/s11083-024-09673-x

Aysan Behnia, Gholam Hossein Fath-Tabar, Gyula O. H. Katona

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摘要

循环正集由元素 1、2、...、n 的循环排列的区间组成，按包容度排序。假设 F 是这样一个区间集合，其中没有一个区间小于其他 s 个区间。F 的最大大小就是在这个条件下确定的。同时也证明了，如果已知在这个集合中一个集合的最大大小不包含一个小的子集合 P，那么在由成对 \((i,j) (1\le i,j\le n)\)组成并按坐标排序的网格集合中，它也能解决同样的问题，且不超过一个加常数。

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Forbidden Subposets in the Cycle Poset

The cycle poset consists of the intervals of the cyclic permutation of the elements 1, 2, ..., n, ordered by inclusion. Suppose that F is a set of such intervals, none of them is a less than s others. The maximum size of F is determined under this condition. It is also shown that if the largest size of a set in this poset without containing a small subposet P is known, it solves the same problem, up to an additive constant, in the grid poset consisting of the pairs \((i,j) (1\le i,j\le n)\) and ordered coordinate-wise.

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