Minimizing the regularity of maximal regular antichains of 2- and 3-sets

TL;DR

This paper investigates the minimal regularity of maximal antichains composed of 2- and 3-element subsets of an n-element set, providing lower bounds and construction methods for such structures.

Contribution

It establishes lower bounds on the regularity parameter and introduces new constructions for regular maximal antichains with minimal regularity.

Findings

Derived lower bounds on the regularity r.

Presented constructions achieving small regularity.

Analyzed properties of maximal antichains of 2- and 3-sets.

Abstract

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.