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TRANSFERENCE FOR THE ERDŐS–KO–RADO THEOREM
Published online by Cambridge University Press: 26 October 2015
Abstract
For natural numbers $n,r\in \mathbb{N}$ with
$n\geqslant r$, the Kneser graph
$K(n,r)$ is the graph on the family of
$r$-element subsets of
$\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of
$K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as
$r/n$ is bounded away from
$1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.
MSC classification
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- Research Article
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- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2015
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