Cycles are strongly Ramsey-unsaturated

We call a graph H 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 in [J. Graph Theory 51 (2006), pp. 22–32], where it is shown that cycles (except for C 4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn , 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 Cn , then r(H)=r(Cn), 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 strongly Ramsey-unsaturated. Our proof uses the regularity method.