Poset saturation of unions of chains

Abstract

A family $\mathcal{G}$ of sets is a(n induced) copy of a poset $P=(P,\leqslant)$ if there exists a bijection $b:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ holds if and only if $b(p)\subseteq b(q)$. The induced saturation number sat$^*(n,P)$ is the minimum size of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain any copy of $P$, but for any $G\in 2^{[n]}\setminus \mathcal{F}$, the family $\mathcal{F}\cup \{G\}$ contains a copy of $P$. We consider sat$^*(n,P)$ for posets $P$ that are formed by pairwise incomparable chains, i.e. $P=\bigoplus_{j=1}^mC_{i_j}$. We make the following two conjectures: (i) sat$^*(n,P)=O(n)$ for all such posets and (ii) sat$^*(n,P)=O(1)$ if not all chains are of the same size. (The second conjecture is known to hold if there is a unique longest among the chains.) We verify these conjectures in some special cases: we prove (i) if all chains are of the same length, we prove (ii) in the first unknown general case: for posets $2C_k+C_1$. Finally, we give an infinite number of examples showing that (ii) is not a necessary condition for sat$^*(n,P)=O(1)$ among posets $P=\bigoplus_{j=1}^mC_{i_j}$: we prove sat$^*(n,(\binom{2t}{t}+1)C_2)=O(1)$ for all $t\ge 1$.