Coloring triangle-free graphs with fixed size

Abstract Combining recent results on colorings and Ramsey theory, we show that if G is a triangle-free graph with e edges then the chromatic number of G is at most ce 1/3 ( log e) −2/3 for some constant c. In a previous paper, we found an upper bound on the chromatic number of a triangle-free graph of genus g. Using the new result, we slightly improve this bound to cg 1/3 ( log g) −2/3 . Both bounds are best possible, up to a constant multiple.