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arXiv · 1206.3007

Maximal antichains of minimum size

Thomas Kalinowski·Uwe Leck·Ian T. Roberts
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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.

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Thomas Kalinowski, Uwe Leck, Ian T. Roberts. 2013-03-09. Maximal antichains of minimum size. https://arxiv.org/abs/1206.3007

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