Induced Ramsey problems for trees and graphs with bounded treewidth

The induced q-color size-Ramsey number r̂ind(H;q) of a graph H is the minimal number of edges a host graph G can have so that every q-edge-coloring of G contains a monochromatic copy of H which is an induced subgraph of G. A natural question, which in the non-induced case has a very long history, asks which families of graphs H have induced Ramsey numbers that are linear in $|H|$. We prove that for every $k,w,q$, if H is an n-vertex graph with maximum degree k and treewidth at most w, then r̂ind(H;q) = Ok,w,q(n). This extends several old and recent results in Ramsey theory. Our proof is quite simple and relies upon a novel reduction argument.