## 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 Erdős, Ko and Rado from 1961 says that, provided *n* ≥ 2*k*, any such system has size at most \((_{k - 1}^{n - 1})\). 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.

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## Additional information

Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.

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Das, S., Gan, W. & Sudakov, B. The minimum number of disjoint pairs in set systems and related problems.
*Combinatorica* **36, **623–660 (2016). https://doi.org/10.1007/s00493-014-3133-0

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### Mathematics Subject Classification (2010)

- 05D05
- 05C65