Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact Ramsey numbers of odd cycles via nonlinear optimisation   For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer
$N$ such that every $k$-colouring of the edges of the complete graph $K_N$
contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$
vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large,
\[ R_k(C_n)=2^{k-1}(n-1)+1. \] This resolves a conjecture of Bondy and
Erd\H{o}s [J. Combin. Th. Ser. B \textbf{14} (1973), 46--54] for large $n$. The
proof is analytic in nature, the first step of which is to use the regularity
method to relate this problem in Ramsey theory to one in nonlinear
optimisation. This allows us to prove a stability-type generalisation of the
above and establish a surprising correspondence between extremal $k$-colourings
for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.

32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys