We call a graph $H$ \emph{Ramsey-unsaturated} if there is an edge in the
complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not
change upon adding it to $H$. This notion was introduced by Balister,
Lehel and Schelp who also showed that cycles (except for $C_4$) are
Ramsey-unsaturated, and conjectured that, moreover, one may add {\em
any} chord without changing the Ramsey number of the cycle $C_n$, unless
$n$ is even and adding the chord creates an odd cycle.
We prove this conjecture for large cycles by showing a stronger
statement: If a graph $H$ is obtained by adding a linear number of
chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum
degree of $H$ is bounded, $H$ is either bipartite (for even $n$) or
almost bipartite (for odd $n$), and $n$ is large.
This motivates us to call cycles \emph{strongly} Ramsey-unsaturated.
Our proof uses the regularity method.