Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Density Tur\'an problem   Let $H$ be a graph on $n$ vertices and let the blow-up graph
  $G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a
cluster
  $A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$
was
  an edge of the graph $H$. As usual, we define the edge density between $A_i$
and $A_j$ as $d(A_i,A_j)=\frac{e(A_i,A_j)}{|A_i||A_j|}.$ We study the following
problem. Given densities $\gamma_{ij}$ for each edge $(i,j)\in E(H)$. Then one
has to decide whether there exists a blow-up graph $G[H]$ with edge densities
at least $\gamma_{ij}$ such that one cannot choose a vertex from each cluster
so that the obtained graph is isomorphic to $H$, i.e, no $H$ appears as a
transversal in $G[H]$. We call $d_{crit}(H)$ the maximal value for which there
exists a blow-up graph $G[H]$ with edge densities $d(A_i,A_j)=d_{crit}(H)$
$((v_i,v_j)\in E(H))$ not containing $H$ in the above sense. Our main goal is
to determine the critical edge density and to characterize the extremal graphs.

8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol