Strengthening Rödl's theorem

Abstract

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 ε>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 ε>0, every H-free graph can be partitioned into a bounded number of subsets inducing ε-restricted graphs.