Sharp threshold for the Erdős–Ko–Rado theorem

For positive integers n$$ n $$ and k$$ k $$ with n≥2k+1$$ n\ge 2k+1 $$, the Kneser graph K(n,k)$$ K\left(n,k\right) $$ is the graph with vertex set consisting of all k$$ k $$‐sets of {1,…,n}$$ \left\{1,\dots, n\right\} $$, where two k$$ k $$‐sets are adjacent exactly when they are disjoint. The independent sets of K(n,k)$$ K\left(n,k\right) $$ are k$$ k $$‐uniform intersecting families, and hence the maximum size independent sets are given by the Erdős–Ko–Rado Theorem. Let Kp(n,k)$$ {K}_p\left(n,k\right) $$ be a random spanning subgraph of K(n,k)$$ K\left(n,k\right) $$ where each edge is included independently with probability p$$ p $$. Bollobás, Narayanan, and Raigorodskii asked for what p$$ p $$ does Kp(n,k)$$ {K}_p\left(n,k\right) $$ have the same independence number as K(n,k)$$ K\left(n,k\right) $$ with high probability. For n=2k+1$$ n=2k+1 $$, we prove a hitting time result, which gives a sharp threshold for this problem at p=3/4$$ p=3/4 $$. Additionally, completing work of Das and Tran and work of Devlin and Kahn, we determine a sharp threshold function for all n>2k+1$$ n>2k+1 $$.