Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small dense subgraphs of polarity graphs and the extremal number for the
  4-cycle   In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is
a polarity graph of a projective plane of order $q$ that has an oval, then $G$
contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O
( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known
lower bounds on the Tur\'{a}n number $\mathrm{ex}(n, C_4)$ for certain values
of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate
concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.

13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin