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 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 K ℓ -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 c ℓ 4 log ℓ has a grid-like-minor of order ℓ . As an application of this result, we prove that the Cartesian product G □ K 2 contains a K ℓ -minor whenever G has treewidth at least c ℓ 4 log ℓ .