Partition-free families of sets

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 m+1 ) + . . . + ( n 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.