  The induced size-Ramsey number $\hat{r}_\text{ind}^k(H)$ of a graph $H$ is
the smallest number of edges a (host) graph $G$ can have such that for any
$k$-coloring of its edges, there exists a monochromatic copy of $H$ which is an
induced subgraph of $G$. In 1995, in their seminal paper, Haxell, Kohayakawa
and Luczak showed that for cycles, these numbers are linear for any constant
number of colours, i.e., $\hat{r}_\text{ind}^k(C_n)\leq Cn$ for some $C=C(k)$.
The constant $C$ comes from the use of the regularity lemma, and has a tower
type dependence on $k$. In this paper we significantly improve these bounds,
showing that $\hat{r}_\text{ind}^k(C_n)\leq O(k^{102})n$ when $n$ is even, thus
obtaining only a polynomial dependence of $C$ on $k$. We also prove
$\hat{r}_\text{ind}^k(C_n)\leq e^{O(k\log k)}n$ for odd $n$, which almost
matches the lower bound of $e^{\Omega(k)}n$. Finally, we show that the ordinary
(non-induced) size-Ramsey number satisfies $\hat{r}^k(C_n)=e^{O(k)}n$ for odd
$n$. This substantially improves the best previous result of $e^{O(k^2)}n$, and
is best possible, up to the implied constant in the exponent. To achieve our
results, we present a new host graph construction which, roughly speaking,
reduces our task to finding a cycle of approximate given length in a graph with
local sparsity.
