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

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdős–Ko–Rado from 1961 says that, provided n ≥ 2k, any such system has size at most ( n−1 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 Bollobás and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdős–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.