  We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced
subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two
well-known graph invariants, the chromatic and clique numbers, can provide
obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph,
the 'nested complexity length', which provides a novel obstruction to
$\mathcal{P}_{g,p}$. For the curve graph this invariant captures the
topological complexity of the surface in graph-theoretic terms; indeed we show
that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the
surface. As a consequence we show that large half-graphs do not have
$\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs
which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We
also reinterpret our obstruction in terms of the first-order theory of the
curve graph, and in terms of RAAG subgroups of the mapping class group
(following Kim and Koberda). Finally, we show that large multipartite subgraphs
cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of
the curve graph, and to conclude that clique size, chromatic number, and nested
complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.
