  In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst
all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal
together with Shearer's classical upper bound for the off-diagonal Ramsey
number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log
n}$.
  We improve this bound by a factor $\sqrt{2}$, as well as obtaining an
analogous bound on the list chromatic number which is tight up to a constant
factor. A bound in terms of the number of edges that is similarly tight
follows, and these results confirm a conjecture of Cames van Batenburg, de
Joannis de Verclos, Kang, and Pirot.
