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

TL;DR

This paper identifies a sharp threshold for the number of variables needed in first-order logic to distinguish between two large random graphs, revealing a precise probabilistic boundary.

Contribution

It establishes the exact asymptotic behavior of the minimum variable count for first-order distinguishability in random graphs, pinpointing a narrow threshold window.

Findings

Threshold for variable count is within {h, h+1, h+2, h+3} with high probability.

Minimum variable count for distinguishability is within {h, h+1, h+2} with high probability.

The probability of the threshold being outside these sets is negligible as n grows.

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 $1-o(1)$.