On coloring graphs with locally small chromatic number

In 1973, P. Erd6s conjectured tha t for each k ~ 2 , there exists a cons tan t ck so that if G is a g raph on n vertices and G has no odd cycle with length less than ckn ~/k, then the chromat ic n u m b e r o f G is at mos t k + 1. Cons t ruc t ions due to LovAsz and Schriver show that ck, if it exists, mus t be at least 1. In this paper we settle Erd6s ' conjecture in the affirmative. We actually prove a s t ronger result which provides an uppe r b o u n d on the chromat ic n u m b e r of a g raph in which we have a bound on the ch romat i c n u m b e r of subg raphs with small diameter .