The size-Ramsey number of powers of bounded degree trees

Given a positive integer (Formula presented.), the (Formula presented.) -colour size-Ramsey number of a graph (Formula presented.) is the smallest integer (Formula presented.) such that there exists a graph (Formula presented.) with (Formula presented.) edges with the property that, in any colouring of (Formula presented.) with (Formula presented.) colours, there is a monochromatic copy of (Formula presented.). We prove that, for any positive integers (Formula presented.) and (Formula presented.), the (Formula presented.) -colour size-Ramsey number of the (Formula presented.) th power of any (Formula presented.) -vertex bounded degree tree is linear in (Formula presented.). As a corollary, we obtain that the (Formula presented.) -colour size-Ramsey number of (Formula presented.) -vertex graphs with bounded treewidth and bounded degree is linear in (Formula presented.), which answers a question raised by Kamčev, Liebenau, Wood and Yepremyan.