  Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and
clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil
(1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon
\leq 1/2$. It was shown by King and Reed that, provided $\Delta$ is large
enough, the conjecture holds for $\varepsilon \leq 1/130,000$. In this article,
we show that the same statement holds for $\varepsilon \leq 1/26$, thus making
a significant step towards Reed's conjecture. We derive this result from a
general technique to bound the chromatic number of a graph where no vertex has
many edges in its neighbourhood. Our improvements to this method also lead to
improved bounds on the strong chromatic index of general graphs. We prove that
$\chi'_s(G)\leq 1.835 \Delta(G)^2$ provided $\Delta(G)$ is large enough.
