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arXiv · 1908.01744

On $L$-close Sperner systems

Daniel Nagy·Balazs Patkos
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Abstract

For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$ belongs to $L$. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of $L$-close Sperner set systems for $L=\{1\}$ and generalize to $|L|=1$ and obtain slightly weaker bounds for arbitrary $L$. We also consider the problem when $L$ might include 0 and reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set systems with all skew distances belonging to $L=\{0,1\}$.

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Daniel Nagy, Balazs Patkos. 2019-08-05. On $L$-close Sperner systems. https://arxiv.org/abs/1908.01744

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