  A proper edge colouring of a graph is adjacent vertex distinguishing if no
two adjacent vertices see the same set of colours. Using a clever application
of the Local Lemma, Hatami (2005) proved that every graph with maximum degree
$\Delta$ and no isolated edge has an adjacent vertex distinguishing edge
colouring with $\Delta + 300$ colours, provided $\Delta$ is large enough. We
show that this bound can be reduced to $\Delta + 19$. This is motivated by the
conjecture of Zhang, Liu, and Wang (2002) that $\Delta + 2$ colours are enough
for $\Delta \geq 3$.
