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For positive integersn   $$ n $$    andk   $$ k $$    withn  ≥  2  k  +  1   $$ n\ge 2k+1 $$   , the Kneser graphK  (  n  ,  k  )   $$ K\left(n,k\right) $$    is the graph with vertex set consisting of allk   $$ k $$   ‐sets of{  1  ,  …  ,  n  }   $$ \left\{1,\dots, n\right\} $$   , where twok   $$ k $$   ‐sets are adjacent exactly when they are disjoint. The independent sets ofK  (  n  ,  k  )   $$ K\left(n,k\right) $$    arek   $$ k $$   ‐uniform intersecting families, and hence the maximum size independent sets are given by the Erdős–Ko–Rado Theorem. LetK    p    (  n  ,  k  )   $$ {K}_p\left(n,k\right) $$    be a random spanning subgraph ofK  (  n  ,  k  )   $$ K\left(n,k\right) $$    where each edge is included independently with probabilityp   $$ p $$   . Bollobás, Narayanan, and Raigorodskii asked for whatp   $$ p $$    doesK    p    (  n  ,  k  )   $$ {K}_p\left(n,k\right) $$    have the same independence number asK  (  n  ,  k  )   $$ K\left(n,k\right) $$    with high probability. Forn  =  2  k  +  1   $$ n=2k+1 $$   , we prove a hitting time result, which gives a sharp threshold for this problem atp  =  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 alln  >  2  k  +  1   $$ n>2k+1 $$   .

Sharp threshold for the Erdős–Ko–Rado theorem