  An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a
simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent
vertices meets the same set of colors. We prove that every graph with maximum
degree $\Delta$ and with no isolated edges has an \avd-coloring with at most
$\Delta+300$ colors, provided that $\Delta >10^{20}$.
