Projective and External Saturation Problem for Posets

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 finite subsets of \(\mathbb {N}\), not necessarily from \(\mathcal {B}_n\), is 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 \(F\cap [n]\ne F'\cap [n]\). Ordinary strongly P-saturated families, i.e., families \(\mathcal {F}\subseteq \mathcal {B}_n\) 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 \(\text {sat}^{*}(n,P)\) and the projective saturation number \(\top \hspace{-10pt}\top \text {sat}(n,P)\), the size of the smallest projective P-saturated family. Note that the projective saturation number might differ for a poset and its dual. Motivated by this, we introduce an even more relaxed and symmetric version of poset saturation, external saturation. We conjecture that all finite posets have bounded external saturation number, and prove this in some special cases.