The Weisfeiler-Leman (WL) dimension is a standard measure in descriptive complexity theory for the structural complexity of a graph. We prove that the WL-dimension of a graph on $n$ vertices is at most $3/20 \cdot n + o(n)= 0.15 \cdot n + o(n)$. The proof develops various techniques to analyze the structure of coherent configurations. This includes sufficient conditions under which a fiber can be restored up to isomorphism if it is removed, a recursive proof exploiting a degree reduction and treewidth bounds, as well as an analysis of interspaces involving small fibers. As a base case, we also analyze the dimension of coherent configurations with small fiber size and thereby graphs with small color class size.

翻译：魏斯费勒-莱曼（WL）维数是描述复杂性理论中衡量图结构复杂性的标准指标。我们证明了$n$个顶点上的图的WL维数至多为$3/20 \cdot n + o(n)= 0.15 \cdot n + o(n)$。该证明开发了多种分析相干配置结构的技术。这些技术包括：在移除一条纤维后能够恢复其同构性的充分条件、利用度约简和树宽界进行递归证明的方法，以及涉及小纤维的间隙分析。作为基础情形，我们还分析了具有小纤维尺寸的相干配置的维数，进而分析了具有小颜色类别尺寸的图的维数。