Saturation for Small Antichains

Abstract

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by ${\text{sat}}^{\ast }(n,{\mathcal{A}}_k)$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that ${\text{sat}}^{\ast }(n,{\mathcal{A}}_k)=(k-1)n(1+o(1))$, and proved this for $k\le 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for ${\text{sat}}^{\ast }(n,{\mathcal{A}}_5)$ and ${\text{sat}}^{\ast }(n,{\mathcal{A}}_6)$. We also give some open problems inspired by our analysis.

8 pages

Cited by in corpus (1)

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