Author:
Bastide Paul,Groenland Carla,Jacob Hugo,Johnston Tom
Abstract
For given positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$, but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use sat$^*(n, k)$ to denote the smallest size of such a family. For all $k$ and sufficiently large $n$, we determine the exact value of sat$^*(n, k)$. Our result implies that sat$^*(n, k)=n(k-1)-\Theta(k\log k)$, which confirms several conjectures on antichain saturation. Previously, exact values for sat$^*(n,k)$ were only known for $k$ up to $6$. We also prove a strengthening of a result of Lehman-Ron which may be of independent interest. We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). The complete version of the paper can be found here \cite{antichainsaturation}.
Reference22 articles.
1. Martin Aigner. Lexicographic matching in Boolean algebras. Journal of Combinatorial Theory, Series B, 14:187-194, 1973.
2. Irina Ðanković and Maria-Romina Ivan. Saturation for small antichains. ArXiv preprint arXiv:2205.07392, 2022.
3. Béla Bajnok and Shahriar Shahriari. Long symmetric chains in the boolean lattice. Journal of Combinatorial Theory, Series A, 75(1):44-54, 1996.
4. Paul Bastide, Carla Groenland, Hugo Jacob, and Tom Johnston. Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron. https://arxiv.org/abs/2207.07391.
5. Robert P. Dilworth. A decomposition theorem for partially ordered sets. Annals of Mathematics, pages 161-166, 1950.
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