On graphs with small subgraphs of large chromatic number

Bollobás, Erdös, Simonovits, and Szemerédi conjectured 1] that for each positive constantc there exists a constantg(c) such that ifG is any graph which cannot be made 3-chromatic by the omission ofcn ² edges, thenG contains a 4-chromatic subgraph with at mostg(c) vertices. Here we establish the following generalization which was suggested by Erdös 2]: For each positive constantc and positive integerk there exist positive integersf _k(c) andn _o such that ifG is any graph with more thann _o vertices having the property that the chromatic number ofG cannot be made less thank by the omission of at mostcn ² edges, thenG contains ak-chromatic subgraph with at mostf _k(c) vertices.