Saturating Sperner families

TL;DR

This paper investigates the minimal sizes of families of sets that saturate the Sperner and related properties, focusing on monotone decreasing properties and specific set sizes.

Contribution

It introduces new bounds and characterizations for the minimal saturating families related to the Sperner property and its variants.

Findings

Derived bounds for minimal saturating families

Characterized saturating families with sets of sizes l and l+1

Extended understanding of Sperner saturation in combinatorics

Abstract

A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the $k$-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of $l$-sets and $(l+1)$-sets.