  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.
