Projective and external saturation problem for posets

Abstract

We introduce two variants of the poset saturation problem. For a poset $P$ and the Boolean lattice $\mathcal{B}_n$, a family $\mathcal{F}$ of sets, not necessarily from $\mathcal{B}_n$, is \textit{projective $P$-saturated} if (i) it does not contain any strong copies of $P$, (ii) for any $G\in \mathcal{B}_n\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a strong copy of $P$, and (iii) for any two different $F,F'\in\mathcal{F}$ we have $F\cap[n]\neq F'\cap [n]$. Ordinary strongly $P$-saturated families, i.e., subfamilies $\mathcal{F}$ required to be from $\mathcal{B}_n$ satisfying (i) and (ii), automatically satisfy (iii) as they lie within $\mathcal{B}_n$. We study what phenomena are valid both for the ordinary saturation number $\mathrm{sat}^*(n,P)$ and the projective saturation number, the size of the smallest projective $P$-saturated family. Note that the projective saturation number might differ for a poset and its dual. We also introduce an even more relaxed and symmetric version of poset saturation, \textit{external saturation}. We conjecture that all finite posets have bounded external saturation number, and prove this in some special cases.