Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On some multicolour Ramsey properties of random graphs   The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer
$m$ such that there exists a graph $G$ on $m$ edges with the property that any
colouring of the edges of $G$ with two colours yields a monochromatic copy of
$F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on
$n$ vertices. In particular, we show that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$
for $n$ sufficiently large. (The upper bound uses expansion properties of
random $d$-regular graphs.) This improves the previous lower bound,
$\hat{R}(P_n) \ge (1+\sqrt{2})n-O(1)$, due to Bollob\'as, and the upper bound,
$\hat{R}(P_n) \le 91n$, due to Letzter. Next we study long monochromatic paths
in edge-coloured random graph $G(n,p)$ with $pn \to \infty$. Let $\alpha > 0$
be an arbitrarily small constant. Recently, Letzter showed that a.a.s.\ any
$2$-edge colouring of $G(n,p)$ yields a monochromatic path of length
$(2/3-\alpha)n$, which is optimal. Extending this result, we show that a.a.s.\
any $3$-edge colouring of $G(n,p)$ yields a monochromatic path of length
$(1/2-\alpha)n$, which is also optimal. In general, we prove that for $r\ge 4$
a.a.s.\ any $r$-edge colouring of $G(n,p)$ yields a monochromatic path of
length $(1/r-\alpha)n$. We also consider a related problem and show that for
any $r \ge 2$, a.a.s.\ any $r$-edge colouring of $G(n,p)$ yields a
monochromatic connected subgraph on $(1/(r-1)-\alpha)n$ vertices, which is also
tight.

37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn