The Saturation Number for the Diamond is Linear

TL;DR

This paper proves that the minimum size of a family of sets avoiding an induced diamond poset but becoming one upon adding any set grows linearly with n, resolving a longstanding gap between known bounds.

Contribution

It establishes a linear lower bound for the saturation number of the diamond poset, improving upon previous sublinear bounds and confirming linear growth.

Findings

The saturation number for the diamond is at least (n+1)/5.

The saturation number for the diamond is at most n+1.

The proof involves a new result on pairs of set systems.

Abstract

For a fixed poset $\mathcal P$ we say that a family $\mathcal F\subseteq\mathcal P([n])$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add a new set to $\mathcal F$, we form an induced copy of $\mathcal P$. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$.\par For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), while it is easy to see that the saturation number is at most $n+1$, the best known lower bound has stayed at $O(\sqrt n)$ since the introduction of the area of poset saturation. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq \frac{n+1}{5}$, establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems.