Maximal antichains of minimum size

TL;DR

This paper investigates the minimal size of maximal antichains of subsets of [n] with sizes in a set K, providing constructions and asymptotic bounds, and connecting the problem to extremal graph theory.

Contribution

The paper introduces a general construction for minimal maximal antichains with set sizes in K, and establishes asymptotic optimality for certain cases, linking the problem to extremal graph theory.

Findings

Construction asymptotically optimal for K containing 2 and 3

Provides bounds for K={2,4}

Connects the problem to the graph removal lemma

Abstract

Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:={1,2,...,n}$. We study the problem to find the smallest possible size of a maximal family $\mathcal{A}$ of subsets of $[n]$ such that $\mathcal{A}$ contains only sets whose size is in $K$, and $A\not\subseteq B$ for all ${A,B}\subseteq\mathcal{A}$, i.e. $\mathcal{A}$ is an antichain. We present a general construction of such antichains for sets $K$ containing 2, but not 1. If $3\in K$ our construction asymptotically yields the smallest possible size of such a family, up to an $o(n^2)$ error. We conjecture our construction to be asymptotically optimal also for $3\not\in K$, and we prove a weaker bound for the case $K={2,4}$. Our asymptotic results are straightforward applications of the graph removal lemma to an equivalent reformulation of the problem in extremal graph theory which is interesting in its own right.