The Induced Saturation Number for $\mathcal{V}_3$ is Linear

TL;DR

This paper establishes that the minimum size of a $ ext{V}_3$-induced saturation family in the Boolean lattice grows linearly with the dimension, providing the first linear lower bound and confirming the order of growth.

Contribution

The paper proves a linear lower bound for the induced saturation number of $ ext{V}_3$, showing it is $ heta(n)$, which was previously unknown.

Findings

$sat^*(n, ext{V}_3) o ext{linear in } n$

Improved lower bound from $2\sqrt{n}$ to $rac{n}{2}$

Confirmed the asymptotic growth as linear for $ ext{V}_3$-induced saturation

Abstract

Given a poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is said to be $\mathcal{P}$-saturated if $\mathcal{F}$ does not contain an induced copy $\mathcal P$, but every proper superset of $\mathcal{F}$ contains one. The minimum size of a $\mathcal P$-saturated family in the $n$-dimensional Boolean lattice is denoted by $sat^*(n,\mathcal{P})$.\par In this paper, we consider the poset $\mathcal V_3$ (the four element poset with one minimal element and three incomparable maximal elements) and show that $sat^*(n,\mathcal{V}_3)\geq \frac{n}{2}$. This represents the first linear lower bound for $sat^*(n,\mathcal{V}_3)$, improving upon the previously best-known bound of $2\sqrt{n}$. Our result establishes that $sat^*(n,\mathcal{V}_3) = \Theta(n)$.