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 @?x@? grid minor is exponential in @?. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A grid-like-minor of order@? in a graph G is a set of paths in G whose intersection graph is bipartite and contains a K"@?-minor. For example, the rows and columns of the @?x@? grid are a grid-like-minor of order @?+1. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least c@?^4log@? has a grid-like-minor of order @?. As an application of this result, we prove that the Cartesian product G@?K"2 contains a K"@?-minor whenever G has treewidth at least c@?^4log@?.

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