Asymptotic behavior of finite permutation groups acting on subsets

We consider primitive subgroups G of the symmetric group Sn, and the sizes of their orbits on subsets X of {1,…,n} as n→∞. We prove a detailed result from which it follows that if the orbits of all large subsets X (with X n/2) are large then G=An or Sn. Our proof invokes the classification of finite simple groups. Questions of this type arise in the study of the threshold behavior of monotone boolean functions.