  In 1943, Hadwiger conjectured that every $K_t$-minor-free graph is
$(t-1)$-colorable for every $t\ge 1$. 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. Very recently,
Norin and Song proved that every graph with no $K_t$ minor is $O(t(\log
t)^{0.354})$-colorable. Improving on the second part of their argument, we
prove that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable
for every $\beta > \frac{1}{4}$.
