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Counting Intersecting and Pairs of Cross-Intersecting Families

Part of: Extremal combinatorics

Published online by Cambridge University Press:  16 October 2017

PETER FRANKL  and
ANDREY KUPAVSKII
PETER FRANKL
Affiliation:
Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Moscow, Russia and Ecole Polytechnique Fédérale de Lausanne, Switzerland (e-mail: kupavskii@yandex.ru)
ANDREY KUPAVSKII
Affiliation:
Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Moscow, Russia and Ecole Polytechnique Fédérale de Lausanne, Switzerland (e-mail: kupavskii@yandex.ru)
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Abstract

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

MSC classification

Primary: 05D05: Extremal set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 1 , January 2018 , pp. 60 - 68
Copyright
Copyright © Cambridge University Press 2017 

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