The size‐Ramsey number of powers of bounded degree trees

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...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2021-06, Vol.103 (4), p.1314-1332
Main Authors: Berger, Sören, Kohayakawa, Yoshiharu, Maesaka, Giulia Satiko, Martins, Taísa, Mendonça, Walner, Mota, Guilherme Oliveira, Parczyk, Olaf
Format: Article
Language:English
Subjects:
05C55 (primary)
05C80
05D10 (secondary)
05D40
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Summary: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 kth 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 Kamčev, Liebenau, Wood and Yepremyan.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms.12408