论文标题
蝴蝶poset的饱和
Saturation for the Butterfly Poset
论文作者
论文摘要
给定有限的poset $ \ mathcal p $，如果$ [n] $ $ \ $ \ MATHCAL p $ - 如果$ \ Mathcal f $包含$ \ Mathcal p $的$ \ Mathcal f $，则称为$ [n] $ $ $ $ $ $ $ $ $ $ \ MATHCAL P $的子集，但将任何其他设置添加到$ \ Mathcal F $创建$ $ p $ p p $ p $。 $ \ MATHCAL P $的饱和数量，由$ \ text {sat}^*（n，\ Mathcal p）$表示，是最小的$ \ Mathcal p $饱和家庭的大小，带有地面套装$ [n] $。在本文中，我们主要对称为The Butterfly的四点Poset感兴趣。 Ferrara，Kay，Kramer，Martin，Reiniger，Smith和Sullivan表明蝴蝶的饱和数在$ \ log_2 {n} $和$ n^2 $之间。我们给出$ n+1 $的线性下限。我们还证明了有关蝴蝶和Poset $ \ Mathcal n $的其他结果。
Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$.
