An even better Density Increment Theorem and its application to Hadwiger's Conjecture
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In 1943, Hadwiger conjectured that every graph with no K_t minor is (t-1)-colorable for every t≥ 1. In the 1980s, Kostochka and Thomason independently proved that every graph with no K_t minor has average degree O(t√(log t)) and hence is O(t√(log t))-colorable. Recently, Norin, Song and the author showed that every graph with no K_t minor is O(t(log t)^β)-colorable for every β > 1/4, making the first improvement on the order of magnitude of the O(t√(log t)) bound. More recently, the author showed that every graph with no K_t minor is O(t (log t)^β)-colorable for every β > 0; more specifically, they are t · 2^ O((loglog 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 (loglog t)^6)-colorable.
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