Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The chromatic number of dense random graphs   The chromatic number $\chi(G)$ of a graph $G$ is defined as the minimum
number of colours required for a vertex colouring where no two adjacent
vertices are coloured the same. The chromatic number of the dense random graph
$G \sim G(n,p)$ where $p \in (0,1)$ is constant has been intensively studied
since the 1970s, and a landmark result by Bollob\'as in 1987 first established
the asymptotic value of $\chi(G)$. Despite several improvements of this result,
the exact value of $\chi(G)$ remains open. In this paper, new upper and lower
bounds for $\chi(G)$ are established. These bounds are the first ones that
match each other up to a term of size $o(1)$ in the denominator: they narrow
down the colouring rate $n/\chi(G)$ of $G \sim G(n,p)$ to an explicit interval
of length $o(1)$, answering a question of Kang and McDiarmid.

22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int