Minimal Diamond-Saturated Families

TL;DR

This paper investigates the minimal size of families of subsets of [n] that are saturated with respect to the diamond poset, establishing new lower bounds and exploring structural properties of such families.

Contribution

It proves a new lower bound of approximately 4√n for the size of diamond-saturated families, improving previous bounds and analyzing their structural characteristics.

Findings

Established that ext{sat}^*(n, ext D_2) ext{ is at least } (4 - o(1)) ext{ } oot n

Provided insights into the properties of small diamond-saturated families

Extended understanding of saturation in poset-related subset families

Abstract

For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$. For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq (4-o(1))\sqrt n$. We also explore the properties that a diamond-saturated family of size $c\sqrt n$, for a constant $c$, would have to have.