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Most Probably Intersecting Families of Subsets

Published online by Cambridge University Press:  02 February 2012

GYULA O. H. KATONA ,
GYULA Y. KATONA  and
ZSOLT KATONA
GYULA O. H. KATONA
Affiliation:
Rényi Institute, Hungarian Academy of Sciences, Budapest Pf 127, 1364Hungary (e-mail: ohkatona@renyi.hu)
GYULA Y. KATONA
Affiliation:
Department of Computer Science, Budapest University of Technology and Economics, Budapest, Magyar tudósok krt. 2, 1117Hungary (e-mail: kiskat@cs.bme.hu)
ZSOLT KATONA
Affiliation:
Haas School of Business, University of California at Berkeley, Berkeley, CA 94720-1900, USA (e-mail: zskatona@haas.berkeley.edu)
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Abstract

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Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all $\lfloor \frac{n}{ 2}\rfloor$-element subsets. We determine the most probably inclusion-free family too, when the number of members is $\binom{n}{ \lfloor \frac{n}{ 2}\rfloor} +1$.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 219 - 227
Copyright
Copyright © Cambridge University Press 2012

References

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