Edges of the uniform random forest of $K_n$ are pairwise negatively correlated for every $n$

Abstract

Let $F_n$ be uniform on all forests of the simple complete graph $K_n$, with isolated vertices allowed. A conjecture of Kahn and of Winkler, studied by Grimmett and Winkler, asserts that any two distinct edges of any finite graph are negatively correlated under the uniform forest measure. Stark proved this for $G=K_n$ once $n$ is sufficiently large, but did not furnish an explicit threshold. We prove it for every $n\geq 2$, strictly whenever two distinct edges exist. The difficulty is concentrated in the disjoint-edge orbit, whose correlation ratio tends to one. We remove this cancellation before estimating anything: the desired inequality becomes an exact comparison among the first two moments of the component count and the expected sum of squared degrees. When the component count fluctuates, this comparison contains a variance term absent from the fixed-component identities of Tang and Zhang. Component marking, tail elimination, and effective Stirling bounds control the three moments for $n\geq 651$; exact integer recurrences cover the remaining values.