Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weighted Turan Problems with Applications   Suppose the edges of $K_n$ are assigned weights by a weight function $w$. We
define the {\em weighted extremal number}
  \[
  \mathrm{ex}(n,w,F):=\max\{w(G)\mid G\subseteq K_n,\text{ and }G\text{ is
}F\text{-free}\}
  \]
  where $w(G):=\sum_{e\in E(G)}w(e)$. In this paper we study this problem for
two types of weights $w$, each of which has an application. The first
application is to an extremal problem in a complete multipartite host graph.
The second application is to the maximum rectilinear crossing number of trees
of diameter 4.

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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int