On "stability" in the Erdös-Ko-Rado Theorem.

Denote by K-p(n, k) the random subgraph of the usual Kneser graph K(n, k) in which edges appear independently, each with probability p. Answering a question of Bollobas, Narayanan, and Raigorodskii, we show that there is a fixed p < 1 such that a.s. (i. e., with probability tending to 1 as k -> infinity) the maximum independent sets of K-p (2k + 1, k) are precisely the sets {A is an element of V (K (2k + 1, k)) : x is an element of A} (x is an element of [2k + 1]). We also complete the determination of the order of magnitude of the "threshold" for the above property for general k and n >= 2 k + 2. This is new for k similar to n/2, while for smaller k it is a recent result of Das and Tran.