What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rödl showed in the 1980s that every H-free graph has large parts that are very sparse or very dense. More precisely, let us say that a graph F on n vertices is ε-restricted if either F or its complement has maximum degree at most εn. Rödl proved that for every graph H, and every $\epsilon >0$, every H-free graph G has a linear-sized set of vertices inducing an ε-restricted graph. We strengthen Rödl's result as follows: for every graph H, and all $\epsilon >0$, every H-free graph can be partitioned into a bounded number of subsets inducing ε-restricted graphs.

对于不包含某个图H的归纳副本的图的结构可以说什么呢？Rödl 在 20 世纪 80 年代表明，每个无H图都有很大的部分非常稀疏或非常密集。更准确地说，如果F或其补集的最大度数至多为εn ，则n 个顶点上的图F是ε 受限的。Rödl 证明了对于每个图H和每个$\epsilon >0$，每个H无图G都有一组线性大小的顶点，从而导出ε限制图。我们按如下方式强化 Rödl 的结果：对于每个图H，并且所有$\epsilon >0$，每个H无图都可以划分为有限数量的子集，从而导出ε受限图。