Polynomial treewidth forces a large grid-like-minor

AbstractRobertson 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 ℓ×ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a Kℓ-minor. For example, the rows and columns of the ℓ×ℓ grid are a grid-like-minor of order ℓ+1. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least cℓ4logℓ has a grid-like-minor of order ℓ. As an application of this result, we prove that the Cartesian product G□K2 contains a Kℓ-minor whenever G has treewidth at least cℓ4logℓ