论文标题
小抗饱和度
Saturation for Small Antichains
论文作者
论文摘要
对于给定的正整数$ k $，我们说$ [n] $的子集的家族是$ k $ - antichain如果不包含$ k $成对无与伦比的套件，但是每当我们添加新的套装时，我们确实会找到$ k $的集合。 $ \ text {sat}^*（n，\ mathcal a_ {k}）$表示最小的此类家族的大小。 Ferrara，Kay，Kramer，Martin，Reiniger，Smith和Sullivan猜想$ \ text {sat}^*（n，\ Mathcal a_ {k {k}）=（k-1）n（k-1）n（1+o（1））$，并以$ k \ leq 4 $ $ k \ leq 4 $证明了这一点。在本文中，我们证明了$ k = 5 $和$ k = 6 $的猜想。此外，我们给出了$ \ text {sat}^*（n，\ mathcal a_5）$和$ \ text {sat}^**（n，\ mathcal a_6）$的确切值。我们还提供了一些受分析启发的开放问题。
For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\text{sat}^*(n, \mathcal A_5)$ and $\text{sat}^*(n, \mathcal A_6)$. We also give some open problems inspired by our analysis.
