  We show that any set of distinct homotopy classes of simple closed curves on
the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k}
\log k)$. Prior to this work, a lemma of Agol, together with the state of the
art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$,
and in fact the assumption of the Riemann hypothesis improved this error term
to the one we obtain $O(\sqrt{k} \log k)$. By contrast, our methods are
elementary, combinatorial, and geometric.
