Balázs Patkós: On $L$-close Sperner systems

For a set $L$ of positive integers, a set system $\cF \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\cF$ 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 determine the order of magnitude of the size of largest set systems with all skew distances belonging to $L_t=\{0,1,\dots,t\}$ and determine the exact maximum size for $L=\{0,1\}$.