高效迫使广义拟随机图成形
Forcing Generalized Quasirandom Graphs Efficiently
AI总结：
本文研究广义拟随机图的结构迫使问题，将同态密度所需测试图的顶点数上界改进至 4q^2−q，并在各部分度数互异时改进至 2q+1。
摘要：
我们研究广义拟随机图，其顶点集由 q 个部分组成（各部分大小未必相同），且每个部分内部以及每对部分之间的边均呈拟随机分布；这类图对应于统计学和网络科学中研究的随机块模型。Lovász 和 Sós 证明，此类图的结构由顶点数至多为 (10q)^q+q 的图的同态密度所迫使；随后，Lovász 改进了论证，表明具有 4(2q+3)^8 个顶点的图即足够。我们的结果表明，对于 q≥2 个部分的广义拟随机图，其结构由顶点数至多为 4q^2−q 的图的同态密度所迫使；并且，如果不同部分中的顶点具有不同的度数，则 2q+1 个顶点即足够。后一结果改进了 Spencer 给出的 8q−4 的界。
英文摘要：
We study generalized quasirandom graphs whose vertex set consists of $q$ parts (of not necessarily the same sizes) with edges within each part and between each pair of parts distributed quasirandomly; such graphs correspond to the stochastic block model studied in statistics and network science. Lovász and Sós showed that the structure of such graphs is forced by homomorphism densities of graphs with at most $(10q)^q+q$ vertices; subsequently, Lovász refined the argument to show that graphs with $4(2q+3)^8$ vertices suffice. Our results imply that the structure of generalized quasirandom graphs with $q\ge 2$ parts is forced by homomorphism densities of graphs with at most $4q^2-q$ vertices, and, if vertices in distinct parts have distinct degrees, then $2q+1$ vertices suffice. The latter improves the bound of $8q-4$ due to Spencer.
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