On L-Close Sperner Systems

For a set L of positive integers, a set system $${\mathcal F}\subseteq 2^{[n]}$$ F ⊆ 2 [ n ] is said to be L-close Sperner, if for any pair F, G of distinct sets in $${\mathcal F}$$ F the skew distance $$sd(F,G)=\min \{|F\setminus G|,|G\setminus F|\}$$ s d ( F , G ) = min { | F \ G | , | G \ F | } belongs to L. We reprove an extremal result of Boros, Gurvich, and Milanič on the maximum size of L-close Sperner set systems for $$L=\{1\}$$ L = { 1 } , generalize it to $$|L|=1$$ | 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, Füredi, and Pach on the size of largest set systems with all skew distances belonging to $$L=\{0,1\}$$ L = { 0 , 1 } .