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 logn 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 Erd˝os, Faudree and Sós. © 2006 Elsevier Inc. All rights reserved.

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Boris Bukh, Benny Sudakov
• Keywords: Graph G, JCT 2007, Largest Homogeneous Subgraph, Short Paper