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 \emph{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.Comment: v2: The bound in the main result has been improved by using the
Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte