Poset saturation of unions of chains
Shengjin Ji, Balázs Patkós, Erfei Yue
TL;DR
The paper investigates the induced poset-saturation number $\mathrm{sat}^*(n,P)$ for posets $P$ formed as disjoint unions of chains, $P=\bigoplus_{j=1}^m C_{i_j}$. It obtains upper bounds via explicit constructions that ensure $P$-freeness and force a copy of $P$ when any outside set is added, with results including a linear bound when all chains have equal length and several constant bounds in cases with mixed chain lengths. Key contributions include a general construction proving $\mathrm{sat}^*(n,mC_k)\le C(m,k)$, a constant-bound result for $\mathrm{sat}^*(n,2C_k+C_1)\le C(k)$, and an infinite family $\mathrm{sat}^*(n,(\binom{2t}{t}+1)C_2)=O(1)$ for all $t\ge 1$, complemented by a discussion of Bollobás-system techniques. These results advance understanding of how the induced saturation number behaves for posets built from chains and illuminate when it can be linear, constant, or fall in between. The methods provide a framework for constructing saturated families and relate to fundamental extremal-set problems in the induced setting.
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 $t\ge 1$.