2017/06/01 by Frankl, Peter, Kupavskii, Andrey
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.1706.00215
Let m(n) denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968 Kleitman proved that m(n) = n\choose m+1+… +n\choose 2m+1 if n=3m+1. Confirming the conjecture of Kleitman, we establish the same equality for the cases n=3m and n=3m+2, and also determine all extremal families. Unlike the case n=3m+1, the extremal families are not unique. This is a plausible reason behind the relative difficulty of our proofs. We completely settle the case of several families as well.