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Probably Intersecting Families are Not Nested

Published online by Cambridge University Press:  09 October 2012

PAUL A. RUSSELL  and
MARK WALTERS
PAUL A. RUSSELL
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected])
MARK WALTERS
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK (e-mail: [email protected])
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Abstract

It is well known that an intersecting family of subsets of an n-element set can contain at most 2n−1 sets. It is natural to wonder how ‘close’ to intersecting a family of size greater than 2n−1 can be. Katona, Katona and Katona introduced the idea of a ‘most probably intersecting family’. Suppose that is a family and that 0 < p < 1. Let (p) be the (random) family formed by selecting each set in independently with probability p. A family is most probably intersecting if it maximizes the probability that (p) is intersecting over all families of size ||.

Katona, Katona and Katona conjectured that there is a nested sequence consisting of most probably intersecting families of every possible size. We show that this conjecture is false for every value of p provided that n is sufficiently large.

Keywords

Primary 05D05
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 146 - 160
Copyright
Copyright © Cambridge University Press 2012

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References

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