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- W4302802823 abstract "Given a set $X$, a collection $mathcal{F}subseteqmathcal{P}(X)$ 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 property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $mathcal{F}subseteqmathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $varepsilon>0$ such that for every $k$ and $|X| geq n_0(k)$ there exists a saturated $k$-Sperner system $mathcal{F}subseteqmathcal{P}(X)$ with cardinality at most $2^{(1-varepsilon)k}$. A collection $mathcal{F}subseteq mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $Sinmathcal{P}(X)setminusmathcal{F}$, $mathcal{F}cup{S}$ contains more chains of length $k+1$ than $mathcal{F}$. Gerbner et al. proved that, if $|X|geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $Oleft(frac{log{k}}{k}2^kright)$ elements. We show that if $|X|geq k^2+k$, then the lower bound is best possible, up to a polynomial factor." @default.
- W4302802823 created "2022-10-07" @default.
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- W4302802823 date "2014-02-23" @default.
- W4302802823 modified "2023-09-28" @default.
- W4302802823 title "On Saturated $k$-Sperner Systems" @default.
- W4302802823 doi "https://doi.org/10.48550/arxiv.1402.5646" @default.
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