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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 Kamčev, Liebenau, Wood and Yepremyan.

The size‐Ramsey number of powers of bounded degree trees