发表机构
École normale supérieure, Université PSL（巴黎高等师范学院，巴黎文理研究大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了Erdős与Gimbel提出的随机图Gₙ的色数与补色数之差是否随n趋于无穷而趋向无穷的问题，得到了符合猜想尺度的下界及分阶段精细化结果，通过带符号共色剖面等方法完成证明。
AI 中文摘要
设ζ(G)表示将图G的顶点集V(G)划分为若干个部分，使得每个部分诱导出一个团或独立集的最小部分数。Erdős和Gimbel提出问题：对于服从G(n,1/2)分布的随机图Gₙ，差χ(Gₙ)-ζ(Gₙ)是否会以高概率随n趋于无穷而趋向无穷？我们沿着n→∞的全序列解决了该问题，并证明：当n→∞时，概率P(χ(Gₙ)-ζ(Gₙ)≥((log2)²/4)log(200/153)·n/(logn)³)→1，这给出了符合猜想的尺度n/(logn)³的下界。我们还得到了分阶段的精细化结果：若δₙ是标准独立数中心的小数部分，则系数可替换为((log2)²A₄(δₙ))/4 - o(1)，其中A₄是显式、连续、非常数的函数，且对所有δ∈[0,1]满足A₄(δ)>log(200/153)。证明使用了支撑在四个连续类大小上的带符号共色剖面，在自然类大小截断的跳变点间保持一致性；通过精确的带符号重叠恒等式将局部单元奖励与二元循环空间因子分离；经高单元与 capped 剩余匹配的规范分解、端点表比较及剩余偶边集的单射限制，得到所需的二阶矩界；再通过有界差论证将所得的稀有带符号见证放大为高概率共色。
英文摘要
Let $ζ(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $χ(G_n)-ζ(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove that $\mathbb P(χ(G_n)-ζ(G_n)\ge ((\log 2)^2/4)\log(200/153)\,n/(\log n)^3)\to1$. This gives a lower bound at the conjectured scale $n/(\log n)^3$. We also obtain a phase-resolved refinement: if $δ_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\log 2)^2A_4(δ_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(δ)>\log(200/153)$ for every $δ\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.
Comments51 pages, 3 figures. Companion Lean 4 formalization and reproducibility record at the cited exact software revision
