TRANSFERENCE FOR THE ERDŐS–KO–RADO THEOREM

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.