Transference for the Erdős-Ko-Rado theorem

J. Balogh, B. Bollobas, B. Narayanan

Forum of Mathematics, Sigma 3, 18 (2015)

#graphtheory #combinatorics #randomgraphs

Two constructions for an intersecting family of r-sets.

For natural numbers, 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 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.