  In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is
$(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason
independently proved that every graph with no $K_t$ minor has average degree
$O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. We show that
every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every
$\beta > 1/4$, making the first improvement on the order of magnitude of the
Kostochka-Thomason bound.
