Maximal Antichains of Minimum Size

Thomas Kalinowski, Uwe Leck, Ian T. Roberts

Abstract


Let $n\geqslant 4$ be a natural number, and let $K$ be a set $K\subseteq [n]:=\{1,2,\dots,n\}$. We study the problem of finding 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.

Keywords


Extremal set theory; Sperner property; maximal antichains; flat antichains

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