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On Erdős–Ko–Rado for random hypergraphs I

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  25 June 2019

A. Hamm  and
J. Kahn
A. Hamm
Affiliation:
Department of Mathematics, Winthrop University, Rock Hill, SC 29733, USA
J. Kahn*
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
*
*Corresponding author. Email: jkahn@math.rutgers.edu
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Abstract

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

MSC classification

Primary: 05D40: Probabilistic methods
Secondary: 05D05: Extremal set theory 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 881 - 916
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Supported by NSF grant DMS1201337.

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