Cycles are strongly Ramsey-unsaturated

Abstract

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 who also proved 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 $C_n$, 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 strongly Ramsey-unsaturated. Our proof uses the regularity method.