Improved Bounds for the Excluded Grid Theorem

We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f : Z → Z, such that for all integers g > 0, every graph of treewidth at least f(g) contains the (g×g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g) = O(g poly log g) is sufficient to ensure the existence of the (g×g)-grid minor in any graph. In this paper we improve this bound to f(g) = O(g poly log g). We introduce a number of new techniques, including a conceptually simple and almost entirely self-contained proof of the theorem that achieves a polynomial bound on f(g).