Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Probabilistic Approach to Problems on Distance Graphs and Graphs of
  Diameters (Candidate-Degree Dissertation Author's Review, in Russian)   The dissertation is related to combinatorial geometry with a strong
probabilistic flavor. The main results can be split into three parts. The
results of the first part guarantee that each "unit distance graph" in the
plane has an induced subgraph with chromatic number at most 4 that covers at
least 91.7 percent of the vertices of the whole graph.
  The results of the second and third parts are related to the standard model
of a random graph with n labeled vertices in which the edges occur
independently with probability p, where p is a function of n. This is known as
the Erdos--Renyi model G(n,p). Given a monotone property of a graph, Erdos and
Renyi's theorem (1960) states that there exists a critical threshold value of
p(n) below which the probability that a random graph has that property tends to
one (as n tends to infinity) and above which the probability tends to zero.
  The main results of the second part are concerned with the (monotone)
property of a graph to be isomorphic to some unit-distance graph in Euclidean
d-space with fixed dimension d. The results of this part guarantee that for d
in {2, 3, 4, 5, 6, 7, 8}, the threshold value of p(n) is big-Theta of 1/n.
Furthermore, the case d = 1 stands apart from the case of higher dimensions;
here the threshold probability is big-Theta of 1/(n^(4/3)).
  The results of the third part are devoted to studying "graphs of diameters"
from the probabilistic standpoint. In particular, it is shown that under some
conditions, almost all graphs of diameters in the plane have chromatic number
less than 3. More generally, it is shown for G(n,p) that graphs of diameters
have a tendency to chromatic degeneration (for large n) when p is close to 0,
but have a tendency to completeness when p is close to 1.

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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: cs