期望阈值处的图分解
Graph Decompositions at the Expectation Threshold
AI总结：
本文证明退化度受限的图可分解为常数块，每块包含阈值与期望阈值同阶，去除了最大度假设，并给出结构扩展和统一分解界。
AI中文摘要：
对于 \\(n\ge3\\) 以及一个顶点数至多为 \\(n\\) 的图 \\(H\\)，令 \\(q(H)\\) 表示其在 \\(G(n,p)\\) 中被包含的期望阈值。我们证明存在绝对常数 \\(a,L>0\\)，使得对于每个 \\(C>0\\)，每个退化度至多为 \\(C\log n/\log\log n\\) 的图都有一个确定性的边分解，分解为至多 \\(\lceil a(C+1)\rceil\\) 块，每一块的普通包含阈值至多为 \\(Lq(H)\\)。这去除了 Ascoli、He、Park 和 Talagrand 在解决 Talagrand 离散凸性问题的方法中所提出的定理中的最大度假设。在常数依赖于固定参数的情况下，一个结构性扩展允许添加 \\(O(\log n)\\) 个具有任意关联边的顶点。统一的主界还给出了每个目标图的一个 \\(O(1+\log\log n)\\) 块分解，并带有绝对阈值乘数。主要成分将 Li 关于扩散测度的双集耦合转化为对指定邻域列表的一个划分，该列表在随机宿主被采样之前就已固定。它允许任意的重叠和重复，并通过在偏置宿主测度下通过共同的双边近似来限制目标需求和宿主供应，从而同时控制所有 Hall 匹配条件。
英文摘要：
For \(n\ge3\) and a graph \(H\) on at most \(n\) vertices, let \(q(H)\) be the expectation threshold for its containment in \(G(n,p)\). We prove that there are absolute constants \(a,L>0\) such that, for every \(C>0\), every graph of degeneracy at most \(C\log n/\log\log n\) has a deterministic edge decomposition into at most \(\lceil a(C+1)\rceil\) pieces, each with ordinary containment threshold at most \(Lq(H)\). This removes the maximum-degree hypothesis from a theorem of Ascoli, He, Park, and Talagrand in their approach to Talagrand's discrete convexity problem. With constants depending on the fixed parameters, a structural extension allows the addition of \(O(\log n)\) vertices with arbitrary incident edges. The uniform main bound also gives an \(O(1+\log\log n)\)-piece decomposition for every target graph, with an absolute threshold multiplier. The main ingredient converts Li's two-set coupling for spread measures into a partition of a prescribed neighborhood list, fixed before the random host is sampled. It allows arbitrary overlaps and repetitions and controls all Hall matching conditions simultaneously by bounding target demand and host supply through common two-sided approximations under the biased host measure.
