In 1943, Hadwiger conjectured that every graph with no $K_t$ minor 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. Recently, Norin, Song and the author showed 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 $O(t\sqrt{\log t})$ bound. More recently, the author showed that every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$-colorable for every $\beta > 0$; more specifically, they are $t \cdot 2^{ O((\log \log t)^{2/3}) }$-colorable. In combination with that work, we show in this paper that every graph with no $K_t$ minor is $O(t (\log \log t)^{6})$-colorable.

中文翻译：

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一个更好的密度增量定理及其在哈德维格猜想中的应用

1943 年，Hadwiger 推测每个没有 $K_t$ 小调的图对于每个 $t\ge 1$ 都是 $(t-1)$-可着色的。在 1980 年代，Kostochka 和 Thomason 独立证明了每个没有 $K_t$ 次要的图都有平均度 $O(t\sqrt{\log t})$，因此是 $O(t\sqrt{\log t})$ -可着色。最近，Norin、Song 和作者表明，对于每一个 $\beta > 1/4$，每个没有 $K_t$ 次要的图都是 $O(t(\log t)^{\beta})$-可着色的，使得$O(t\sqrt{\log t})$ 界限的数量级上的第一次改进。最近，作者展示了每个没有 $K_t$ 次要的图对于每个 $\beta > 0$ 都是 $O(t (\log t)^{\beta})$-colorable; 更具体地说，它们是 $t \cdot 2^{ O((\log \log t)^{2/3}) }$-可着色的。结合这项工作，我们在本文中展示了每个没有 $K_t$ 次要的图都是 $O(t (\log \log t)^{6})$-可着色的。