AI 中文总结
本文证明对任意 $n\geq 2$，完全图 $K_n$ 的一致随机森林中任意两条不同边均严格负相关，解决了 Kahn 和 Winkler 猜想在完全图上的显式阈值问题。
AI 中文摘要
设 $F_n$ 在简单完全图 $K_n$ 的所有森林上均匀分布，允许孤立顶点存在。Kahn 和 Winkler 的一个猜想（由 Grimmett 和 Winkler 研究）断言：在均匀森林测度下，任意有限图的任意两条不同边都是负相关的。Stark 证明了当 $n$ 足够大时该结论对 $G=K_n$ 成立，但未给出显式阈值。我们证明该结论对每个 $n\geq 2$ 都成立，且只要存在两条不同边即为严格负相关。难点集中在不相交边轨道，其相关比趋于 1。我们在估计任何量之前先消除这一抵消：所需不等式转化为分量数前两阶矩与平方度期望和之间的精确比较。当分量数波动时，该比较包含一个方差项，而 Tang 和 Zhang 的固定分量恒等式中没有此项。通过分量标记、尾部消除和有效的 Stirling 界，我们控制了 $n\geq 651$ 时的三个矩；精确整数递推覆盖其余取值。
英文摘要
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.
Comments18 pages. Ancillary files include the exact verification program. Companion repository: https://github.com/agupta/uniform-forests-complete-graphs
