Halfway to Hadwiger's Conjecture

Abstract: 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 $O(t(\log t)<sup>{0.354})$-colorable. Improving on the second part of their argument, we prove that every graph with no $K_t$ minor is $O(t(\log t)<sup>{\beta})$-colorable for every $\beta &gt; \frac{1}{4}$.