Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of Mader's conjecture on large clique subdivisions in $C_4$-free
  graphs   Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such
that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of
a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular,
when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that
every $C_4$-free graph has a subdivision of a clique with order linear in the
average degree of the original graph. In general, the widely conjectured
asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests
our result is tight up to the constant $c(s,t)$.

7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph