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 $f: Z^+\rightarrow Z^+$,suchthatforallintegers$g>0$, every graph of treewidth at least $f(g)$ contains the $(g\times g)$-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 $f(g)=O(g^{98}\mathrm{poly}\log g)$ is sufficient to ensure the existence of the $(g\times g)$-grid minor in any graph. In this paper we improve this bound to $f(g)=O(g^{19}\mathrm{poly}\log g)$. 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 $f(g)$.