Linear Saturation for $\mathcal N$ via Butterflies
Maria-Romina Ivan, Nandi Wang
TL;DR
This work proves that the induced saturation number for the $4$-point poset $\mathcal{N}$ satisfies $sat^*(n, \mathcal{N}) = \Theta(n)$ by establishing a new linear lower bound $sat^*(n, \mathcal{N}) \ge \frac{n+6}{4}$. A central tool is a midpoint lemma: in any $\mathcal{N}$-saturated family, configurations with two antichains sandwiching each other force a 'middle' set $M$ with $\bigcup B_i \subseteq M \subseteq \bigcap A_j$, and for butterflies this midpoint is comparable to all component elements, constraining the structure. The main argument builds a graph on forced singleton flips $(S_i, S_i\cup\{i\})$ that must be acyclic, yielding a lower bound that, together with a trivial $2n$ upper bound, confirms linear growth. The paper also outlines structural implications and open problems regarding the precise constant and full characterization of $\mathcal{N}$-saturated families, and conjectures the exact maximum constant $2n$ for the saturation number.
Abstract
Given a finite poset $\mathcal P$, how small can a family $\mathcal F$ of subsets of $[n]$ be such that $\mathcal F$ does not contain an induced copy of $\mathcal P$, but $\mathcal F\cup\{X\}$ contains such a copy for all $X\in\mathcal P([n])\setminus\mathcal F$? This is known as the induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$. The main conjecture in the area is that the induced saturation number for any poset is either bounded, or linear. In this paper we establish linearity for the induced saturation number of the 4-point poset $\mathcal N$. Previously, it was known that $2\sqrt n\leq\text{sat}^*(n,\mathcal N)\leq 2n$. We show that $\text{sat}^*(n,\mathcal N)\geq\frac{n+6}{4}$. A crucial role in the proof is played by a structural feature of $\mathcal N$-saturated families, namely that if the family contains two antichains, one completely above the other, then it must also contain a `middle' point -- greater than one antichain and less than the other.