Local geometry of the k-curve graph

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Let S be an orientable surface with negative Euler characteristic. For k ∈ N, let Cₖ(S) denote the 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 Teichmüller space and of the mapping class group captures local combinatorial properties of Cₖ(S). Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large k: the clique number of Cₖ(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 Cₖ (quasi-polynomial in k); and the diameter in C₀(S) of a large clique in Cₖ(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.