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Erdős–Ko–Rado in Random Hypergraphs

Published online by Cambridge University Press:  01 September 2009

JÓZSEF BALOGH ,
TOM BOHMAN  and
DHRUV MUBAYI
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: [email protected])
TOM BOHMAN
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA (e-mail: [email protected])
DHRUV MUBAYI
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: [email protected])
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Abstract

Let 3 ≤ k < n/2. We prove the analogue of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph Gk(n, p) when k < (n/2)1/3; that is, we show that with probability tending to 1 as n → ∞, the maximum size of an intersecting subfamily of Gk(n, p) is the size of a maximum trivial family. The analogue of the Erdős–Ko–Rado theorem does not hold for all p when k ≫ n1/3.

We give quite precise results for k < n1/2−ϵ. For larger k we show that the random Erdős–Ko–Rado theorem holds as long as p is not too small, and fails to hold for a wide range of smaller values of p. Along the way, we prove that every non-trivial intersecting k-uniform hypergraph can be covered by k2 − k + 1 pairs, which is sharp as evidenced by projective planes. This improves upon a result of Sanders [7]. Several open questions remain.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 629 - 646
Copyright
Copyright © Cambridge University Press 2009

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