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

Abstract: 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 $\frac{1}{4}-o(1)$ 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 $1-o(1)$.