Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic probabilities of extension properties and random
  $l$-colourable structures   We consider a set $\mbK = \bigcup_{n \in \mbbN}\mbK_n$ of {\em finite}
structures such that all members of $\mbK_n$ have the same universe, the
cardinality of which approaches $\infty$ as $n\to\infty$. Each structure in
$\mbK$ may have a nontrivial underlying pregeometry and on each $\mbK_n$ we
consider a probability measure, either the uniform measure, or what we call the
{\em dimension conditional measure}. The main questions are: What conditions
imply that for every extension axiom $\varphi$, compatible with the defining
properties of $\mbK$, the probability that $\varphi$ is true in a member of
$\mbK_n$ approaches 1 as $n \to \infty$? And what conditions imply that this is
not the case, possibly in the strong sense that the mentioned probability
approaches 0 for some $\varphi$?
  If each $\mbK_n$ is the set of structures with universe ${1, ..., n}$, in a
fixed relational language, in which certain "forbidden" structures cannot be
weakly embedded and $\mbK$ has the disjoint amalgamation property, then there
is a condition (concerning the set of forbidden structures) which, if we
consider the uniform measure, gives a dichotomy; i.e. the condition holds if
and only if the answer to the first question is `yes'. In general, we do not
obtain a dichotomy, but we do obtain a condition guaranteeing that the answer
is `yes' for the first question, as well as a condition guaranteeing that the
answer is `no'; and we give examples showing that in the gap between these
conditions the answer may be either `yes' or `no'. This analysis is made for
both the uniform measure and for the dimension conditional measure. The later
measure has closer relation to random generation of structures and is more
"generous" with respect to satisfiability of extension axioms.

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The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th