🤖 AI Summary
This paper investigates the minimum number $h(n)$ of variables required in first-order logic to distinguish two independent Erdős–Rényi random graphs $G(n,1/2)$. The central question is: what is the smallest $k$ such that a $k$-variable first-order sentence distinguishes the graphs with nontrivial probability—specifically, $frac{1}{4} - o(1)$? Employing combinatorial probabilistic analysis, model-theoretic techniques, and precise characterizations of the expressive power of variable-restricted first-order logic, the authors establish, for the first time, an *extremely sharp threshold* for this distinguishability: the minimal $k$ lies almost surely (with probability $1-o(1)$) within a length-3 interval ${h, h+1, h+2}$—improving upon the prior best-known bound of a length-4 interval. Furthermore, they prove that the optimal distinguishing success probability is asymptotically tight at $1/4$, thereby revealing the fundamental precision limit of first-order logic for expressing structural distinctions on 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 $1-o(1)$.
