Most primitive groups are full automorphism groups of edge-transitive hypergraphs

We prove that, for a primitive permutation group G acting on a set X of size n  , other than the alternating group, the probability that Aut(X,YG)=GAut(X,YG)=G for a random subset Y of X  , tends to 1 as n→∞n→∞. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M.H. Klin. Moreover, we give an upper bound n1/2+ϵn1/2+ϵ for the minimum size of the edges in such a hypergraph. This is essentially best possible.