Further progress towards Hadwiger's conjecture

In 1943, Hadwiger conjectured that every graph with no $Kt$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $Kt$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and the author showed that every graph with no $Kt$ minor is $O(t(\log t)^{{\beta})$-colorable} for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. Building on that work, we show in this paper that every graph with no $Kt$ minor is $O(t (\log t)^{{\beta})$-colorable} for every $\beta > 0$. More specifically in conjunction with another paper by the author, they are $O(t \cdot (\log \log t)^{{18})$-colorable.}