The size-Ramsey number of powers of bounded degree trees

• Main Authors: Berger, Sören, Kohayakawa, Yoshiharu, Maesaka, Giulia Satiko, Martins, Taísa, Mendonça, Walner, Mota, Guilherme Oliveira, Parczyk, Olaf
• Format: Journal Article
• Language: English
• Published: 12-07-2019
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.1907.03466

Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph
$H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$
edges with the property that, in any colouring of $E(G)$ with $s$ colours,
there is a monochromatic copy of $H$. We prove that, for any positive integers
$k$ and $s$, the $s$-colour size-Ramsey number of the $k$th power of any
$n$-vertex bounded degree tree is linear in $n$. As a corollary we obtain that
the $s$-colour size-Ramsey number of $n$-vertex graphs with bounded treewidth
and bounded degree is linear in $n$, which answers a question raised by
Kam\v{c}ev, Liebenau, Wood and Yepremyan [The size Ramsey number of graphs with
bounded treewidth, arXiv:1906.09185 (2019)].