A very sharp threshold for first order logic distinguishability of random graphs

In this paper we find an integer h = h(n) such that the minimum number of variables of a first order sentence that distinguishes between two independent uniformly distributed random graphs of size n with the asymptotically largest possible probability 14 −o(1) belongs to {h,h+1,h+2,h+3}. We also prove that the minimum (random) k such that two independent random graphs are distinguishable by a first order sentence with k variables belongs to {h,h+1,h+2} with probability 1−o(1).