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

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 ≥ 2k, 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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Correspondence to Benny Sudakov.

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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