  Let $S$ be an orientable surface with negative Euler characteristic. For $k
\in \mathbb{N}$, let $\mathcal{C}_{k}(S)$ denote the $\textit{k-curve graph}$,
whose vertices are isotopy classes of essential simple closed curves on $S$,
and whose edges correspond to pairs of curves that can be realized to intersect
at most $k$ times. The theme of this paper is that the geometry of
Teichm\"uller space and of the mapping class group captures local combinatorial
properties of $\mathcal{C}_{k}(S)$. Using techniques for measuring distance in
Teichm\"uller space, we obtain upper bounds on the following three quantities
for large $k$: the clique number of $\mathcal{C}_{k}(S)$ (exponential in $k$,
which improves on all previously known bounds and which is essentially sharp);
the maximum size of the intersection, whenever it is finite, of a pair of links
in $\mathcal{C}_{k}$ (quasi-polynomial in $k$); and the diameter in
$\mathcal{C}_{0}(S)$ of a large clique in $\mathcal{C}_{k}(S)$ (uniformly
bounded). As an application, we obtain quasi-polynomial upper bounds, depending
only on the topology of $S$, on the number of short simple closed geodesics on
any square-tiled surface homeomorphic to $S$.
