On L -Close Sperner Systems
2021
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 Milanic on the maximum size of L-close Sperner set systems for $$L=\{1\}$$
, generalize it 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, Furedi, and Pach on the size of largest set systems with all skew distances belonging to $$L=\{0,1\}$$
.
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