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arXiv · 2402.14113

Saturation of $k$-chains in the Boolean lattice

Ryan R. Martin·Nick Veldt
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Abstract

Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner et al. proved that the smallest saturated $k$-Sperner system contains at least $2^{k/2-1}$ elements, and later, Morrison, Noel, and Scott showed that the smallest such set contains no more than $2^{0.976723k}$ elements. We improve both the upper and lower bounds, showing that the size of the smallest saturated $k$-Sperner system lies between $\sqrt{k}2^{k/2}$ and $2^{0.961471k}$.

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Ryan R. Martin, Nick Veldt. 2024-02-21. Saturation of $k$-chains in the Boolean lattice. https://arxiv.org/abs/2402.14113

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