Extremal triangle-free graphs with chromatic number at least four

Let G be an n-vertex triangle-free graph. The celebrated Mantel's theorem showed that e (G) ²4. In 1962, Erdos (together with Gallai), and independently Andr\'asfai, proved that if G is non-bipartite then e (G) (n-1) ²4+1. In this paper, we extend this result and show that if G has chromatic number at least four and n 150, then e (G) (n-3) ²4+5. The blow-up of Gr\"otzsch graph shows that this bound is best possible.