On coloring graphs with locally small chromatic number

In 1973, P. Erdös conjectured that for each kε2, there exists a constant c _k so that if G is a graph on n vertices and G has no odd cycle with length less than c _k n ^1/k, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c _k, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.