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Probably Intersecting Families are Not Nested

Published online by Cambridge University Press:  09 October 2012

Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK (e-mail:


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.


Copyright © Cambridge University Press 2012

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[1]Ahlswede, R. (1980) Simple hypergraphs with maximal number of adjacent pairs of edges. J. Combin. Theory Ser. B 28 164167.CrossRefGoogle Scholar
[2]Ahlswede, R. and Katona, G. O. H. (1978) Graphs with maximal number of adjacent pairs of edges. Acta Math. Acad. Sci. Hungar. 32 97120.CrossRefGoogle Scholar
[3]Frankl, P. (1977) On the minimum number of disjoint pairs in a family of finite sets. J. Combin. Theory Ser. A 22 249251.CrossRefGoogle Scholar
[4]Katona, G. O. H., Katona, G. Y. and Katona, Z. (2012) Most probably intersecting families of subsets. Combin. Probab. Comput. 21 219227.CrossRefGoogle Scholar
[5]Russell, P. A. (2012) Compressions and probably intersecting families. Combin. Probab. Comput. 21 301313.CrossRefGoogle Scholar
[6]Wagner, S. and Wang, H. (2009) On a problem of Ahlswede and Katona. Studia Sci. Math. Hungar. 46 423435.Google Scholar