Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $elltimesell$ grid minor is exponential in $ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A emph{grid-like-minor of order} $ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{ell}$-minor. For example, the rows and columns of the $elltimesell$ grid are a grid-like-minor of order $ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $cell^4sqrt{logell}$ has a grid-like-minor of order $ell$. As an application of this result, we prove that the cartesian product $Gsquare K_2$ contains a $K_{ell}$-minor whenever $G$ has treewidth at least $cell^4sqrt{logell}$.