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arXiv · 1609.01001

Transference for the Erd\H{o}s-Ko-Rado theorem

József Balogh·Béla Bollobás·Bhargav Narayanan
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Abstract

For natural numbers $n,r \in \mathbb{N}$ with $n\ge r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\dots,n\}$ 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 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 Erd\H{o}s-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.

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József Balogh, Béla Bollobás, Bhargav Narayanan. 2016-09-05. Transference for the Erd\H{o}s-Ko-Rado theorem. https://arxiv.org/abs/1609.01001

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