Title: Partition-free families of sets,
Abstract: 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
$m(n) = {n\choose m+1}+\ldots +{n\choose 2m+1}$ 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.