Extremal triangle-free graphs with chromatic number at least four

Abstract

Let $G$ be an $n$-vertex triangle-free graph. The celebrated Mantel's theorem showed that $e(G)\leq \lfloor\frac{n^2}{4}\rfloor$. In 1962, Erd\H{o}s (together with Gallai), and independently Andr\'{a}sfai, proved that if $G$ is non-bipartite then $e(G)\leq \lfloor\frac{(n-1)^2}{4}\rfloor+1$. In this paper, we extend this result and show that if $G$ has chromatic number at least four and $n\geq 90$, then $e(G)\leq \lfloor\frac{(n-3)^2}{4}\rfloor+5$. The blow-ups of Gr\"{o}tzsch graph shows that this bound is best possible.