Saturation for Small Antichains

TL;DR

This paper proves a conjecture about the minimal size of families of subsets of [n] that are saturated with respect to avoiding k pairwise incomparable sets, confirming it for k=5 and 6, and providing exact values.

Contribution

It extends the known results by proving the conjecture for k=5 and 6, and determines exact values for these cases, advancing understanding of saturated antichain families.

Findings

Confirmed the conjecture for k=5 and 6.

Provided exact values for saturation numbers for k=5 and 6.

Identified open problems for future research.

Abstract

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\text{sat}^*(n, \mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\leq 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\text{sat}^*(n, \mathcal A_5)$ and $\text{sat}^*(n, \mathcal A_6)$. We also give some open problems inspired by our analysis.