Local geometry of the k-curve graph

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 $S$.