# On $L$-close Sperner systems

## TL;DR

This paper studies the maximum size of special set systems called $L$-close Sperner systems, generalizing known results and providing new bounds for various choices of $L$, including cases with zero skew distance.

## Contribution

It rederives extremal bounds for $L$-close Sperner systems and extends these results to broader classes of $L$, including cases with zero skew distance.

## Key findings

- Reproved bounds for $L= ext{\{1\}}$
- Generalized bounds for arbitrary $L$ with $|L|=1$
- Reproved a theorem for $L=\{0,1\}$

## 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 $L=\{0,1\}$.