Partition-free families of sets

TL;DR

This paper determines the maximum size of set families avoiding two disjoint sets with their union for all n, confirming Kleitman's conjecture and characterizing extremal families, with implications for combinatorial set theory.

Contribution

It proves Kleitman's conjecture for all cases of n modulo 3 and characterizes the extremal families, extending previous partial results in combinatorics.

Findings

Confirmed the maximum size formula for all n

Characterized all extremal families for these cases

Showed non-uniqueness of extremal families when n=3m+2

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.