  By the Grid Minor Theorem of Robertson and Seymour, every graph of
sufficiently large tree-width contains a large grid as a minor. Tree-width may
therefore be regarded as a measure of 'grid-likeness' of a graph.
  The grid contains a long cycle on the perimeter, which is the
$\mathbb{F}_2$-sum of the rectangles inside. Moreover, the grid distorts the
metric of the cycle only by a factor of two. We prove that every graph that
resembles the grid in this algebraic sense has large tree-width:
  Let $k, p$ be integers, $\gamma$ a real number and $G$ a graph. Suppose that
$G$ contains a cycle of length at least $2 \gamma p k$ which is the
$\mathbb{F}_2$-sum of cycles of length at most $p$ and whose metric is
distorted by a factor of at most $\gamma$. Then $G$ has tree-width at least
$k$.
