Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An improvement on {\L}uczak's connected matchings method   A connected matching in a graph $G$ is a matching contained in a connected
component of $G$. A well-known method due to {\L}uczak reduces problems about
monochromatic paths and cycles in complete graphs to problems about
monochromatic connected matchings in almost complete graphs. We show that these
can be further reduced to problems about monochromatic connected matchings in
complete graphs.
  We illustrate the potential of this new reduction by showing how it can be
used to determine the $3$-colour Ramsey number of long paths, using a simpler
argument than the original one by Gy\'arf\'as, Ruszink\'o, S\'ark\"ozy, and
Szemer\'edi (2007).

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
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con