Tree-width and planar minors

Robertson and the second author [7] proved in 1986 that for all h   there exists f(h)f(h) such that for every h-vertex simple planar graph H, every graph with no H  -minor has tree-width at most f(h)f(h); but how small can we make f(h)f(h)? The original bound was an iterated exponential tower, but in 1994 with Thomas [9] it was improved to 2O(h5)2O(h5); and in 1999 Diestel, Gorbunov, Jensen, and Thomassen [3] proved a similar bound, with a much simpler proof. Here we show that f(h)=2O(hlog⁡(h))f(h)=2O(hlog⁡(h)) works. Since this paper was submitted for publication, Chekuri and Chuzhoy [2] have announced a proof that in fact f(h)f(h) can be taken to be O(h100)O(h100).