A Polynomial Upper Bound for Poset Saturation

TL;DR

This paper establishes that the induced saturation number for any finite poset grows at most polynomially with the size of the ground set, providing a universal upper bound based on the poset's size.

Contribution

It proves a polynomial upper bound for the induced saturation number of any finite poset, a significant step in understanding poset saturation growth.

Findings

Induced saturation number is polynomially bounded by the size of the ground set.

The bound depends on the size of the poset, specifically at most quadratic in the size.

Bounding VC-dimension is key to establishing the polynomial growth.

Abstract

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that $\text{sat}^*(n, \mathcal P)=O(n^c)$, where $c\leq|\mathcal{P}|^2/4+1$ is a constant depending on $\mathcal P$ only. We obtain this result by bounding the VC-dimension of our family.