The minimum number of disjoint pairs in set systems and related problems

TL;DR

This paper determines the minimum number of disjoint pairs in large k-uniform set systems, confirming conjectures and extending classical theorems like Erdős-Ko-Rado to include disjoint pairs, matchings, and t-disjoint pairs.

Contribution

It provides exact minimum counts of disjoint pairs in small k-uniform families and extends Erdős-Ko-Rado type results to new related problems.

Findings

Confirmed Bollobás and Leader's conjecture for small k.

Determined minimum disjoint pairs in larger set systems.

Extended Erdős-Ko-Rado theorem to matchings and t-disjoint pairs.

Abstract

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \choose k-1}$. A natural question, which was asked by Ahlswede in 1980, is how many disjoint pairs must appear in a set system of larger size. Except for the case k=2, solved by Ahlswede and Katona, this problem has remained open for the last three decades. In this paper, we determine the minimum number of disjoint pairs in small k-uniform families, thus confirming a conjecture of Bollobas and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdos-Ko-Rado theorem, determining the minimum number of matchings of size q and the minimum number of t-disjoint pairs that appear in set systems larger than the corresponding extremal bounds. In the latter case, this provides a partial solution to a problem of Kleitman and West.