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 $m(n)=\left(\genfrac{}{}{0ex}{}{n}{m+1}\right)+\dots +\left(\genfrac{}{}{0ex}{}{n}{2m+1}\right)$ 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.