Sharp threshold for the Erdős-Ko-Rado theorem

Abstract

For positive integers n and k with n 2k + 1, the Kneser graph K(n, k) is the graph with vertex set consisting of all k-sets of {1, . . . , n}, where two k-sets are adjacent exactly when they are disjoint. Let K p (n, k) be a random spanning subgraph of K(n, k) where each edge is included independently with probability p. Bollobás, Narayanan, and Raigorodskii asked for what p does K p (n, k) have the same independence number as K(n, k) with high probability. For n = 2k + 1, we prove a hitting time result, which gives a sharp threshold for this problem at p = 3/4. Additionally, completing work of Das and Tran and work of Devlin and Kahn, we determine a sharp threshold function for all n > 2k + 1.