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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)=nm+1+⋯+n2m+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.