Robertson and the second author [7] proved in 1986 that for all h there exists f(h) such that for every h-vertex simple planar graph H, every graph with no H-minor has treewidth at most f (h); but how small can we make f (h)? The original bound was an iterated exponential tower, but in 1994 with Thomas [9] it was improved to 2(O)(h(5)); and in 1999 Diestel, Gorbunov, Jensen, and Thomassen [3] proved a similar bound, with a much simpler proof. Here we show that f(h) = 2(O)(h log(h)) works. Since this paper was submitted for publication, Chekuri and Chuzhoy [2] have announced a proof that in fact f (h) can be taken to be O(h(100)). (C) 2014 Elsevier Inc. All rights reserved.
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