A very sharp threshold for first order logic distinguishability of random graphs
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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 1/4-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).
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