Induced subgraphs of Ramsey graphs with many distinct degrees

An induced subgraph is called homogeneous if it is either a clique or an independent set. Let hom ( G ) denote the size of the largest homogeneous subgraph of a graph G. In this short paper we study properties of graphs on n vertices with hom ( G ) ⩽ C log n for some constant C. We show that every such graph contains an induced subgraph of order αn in which β n vertices have different degrees, where α and β depend only on C. This proves a conjecture of Erdős, Faudree and Sós.