Authors’ abstract: For a graph \(H\) and an integer \(r\geq 2\), the induced \(r\)-size-Ramsey number of \(H\) is defined to be the smallest integer \(m\) for which there exists a graph \(G\) with \(m\) edges with the following property: however one colours the edges of \(G\) with \(r\) colours, there always exists a monochromatic induced subgraph \(H'\) of \(G\) that is isomorphic to \(H\). This is a concept closely related to the classical \(r\)-size-Ramsey number of Erdös, Faudree, Rousseau and Schelp and to the \(r\)-induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rödl, and Trotter. Here, we prove a result that implies that the induced \(r\)-size-Ramsey number of the cycle \(C^\ell\) is at most \(c_r\ell\) for some constant \(c_r\) that depends only upon \(r\). Thus, we settle a conjecture of Graham and Rödl, which states that the above holds for the path \(P^\ell\) of order \(\ell\), and also generalize in part a result of Bollobás, Burr and Reimer that implies that the \(r\)-size-Ramsey number of the cycle \(C^\ell\) is linear in \(\ell\). Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemerédi.