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

TL;DR

This paper demonstrates that the independence number of a randomly edge-deleted Kneser graph remains equal to that of the original graph under certain conditions, extending the Erd ext{"o}s-Ko-Rado theorem to a probabilistic setting.

Contribution

It establishes a probabilistic stability result for the Erd ext{"o}s-Ko-Rado theorem in random subgraphs of Kneser graphs, with new combinatorial estimates.

Findings

Independence number preserved with high probability when edges are randomly deleted

Results hold for edge retention probabilities that are quite small

Provides new bounds on disjoint pairs in families based on their distance from intersecting families

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.