Transference for the ErdH os-Ko-Rado theorem

For natural numbers n, ∈ 2 ℕ with n ≥ r, the Kneser graph K (n, r) is the graph on the family of r-element subsets of {1; ⋯, 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 Erdos-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.