Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Limits of Order Types   We apply ideas from the theory of limits of dense combinatorial structures to
study order types, which are combinatorial encodings of finite point sets.
Using flag algebras we obtain new numerical results on the Erd\H{o}s problem of
finding the minimal density of 5-or 6-tuples in convex position in an arbitrary
point set, and also an inequality expressing the difficulty of sampling order
types uniformly. Next we establish results on the analytic representation of
limits of order types by planar measures. Our main result is a rigidity
theorem: we show that if sampling two measures induce the same probability
distribution on order types, then these measures are projectively equivalent
provided the support of at least one of them has non-empty interior. We also
show that some condition on the Hausdorff dimension of the support is necessary
to obtain projective rigidity and we construct limits of order types that
cannot be represented by a planar measure. Returning to combinatorial geometry
we relate the regularity of this analytic representation to the aforementioned
problem of Erd\H{o}s on the density of k-tuples in convex position, for large
k.

35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat