Let
S
S
be an orientable surface with negative Euler characteristic. For
k
∈
N
k \in \mathbb {N}
, let
C
k
(
S
)
\mathcal {C}_{k}(S)
denote the k-curve graph, whose vertices are isotopy classes of essential simple closed curves on
S
S
and whose edges correspond to pairs of curves that can be realized to intersect at most
k
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
C
k
(
S
)
\mathcal {C}_{k}(S)
, for large
k
k
. Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large
k
k
: the clique number of
C
k
(
S
)
\mathcal {C}_{k}(S)
(exponential in
k
k
, which improves on previous bounds of Juvan, Malnič, and Mobar and Przytycki); the maximum size of the intersection, whenever it is finite, of a pair of links in
C
k
\mathcal {C}_{k}
(quasi-polynomial in
k
k
); and the diameter in
C
0
(
S
)
\mathcal {C}_{0}(S)
of a large clique of
C
k
(
S
)
\mathcal {C}_{k}(S)
(uniformly bounded). As an application, we obtain quasi-polynomial upper bounds, depending only on the topology of
S
S
, on the number of short simple closed geodesics on any unit-square tiled surface homeomorphic to
S
S
.