Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the complexity of finite subgraphs of the curve graph   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 $\mathcal{P}_{g,p}$.

34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
16
The categories of this paper are: stat
26
The categories of this paper are: bayes-an