A note on the monotonicity of mixed Ramsey numbers

For two graphs, GG and HH, an edge coloring of a complete graph is (G,H)(G,H)-good if there is no monochromatic subgraph isomorphic to GG and no rainbow subgraph isomorphic to HH in this coloring. The set of numbers of colors used by (G,H)(G,H)-good colorings of KnKn is called a mixed Ramsey spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It is shown that the answer is “yes” if GG is not a star and HH does not contain a pendant edge.

► The set of numbers of colors in mixed Ramsey colorings, called a spectrum, is studied. ► It is shown that the spectrum is an interval for a large class of graphs. ► An example is found for which the spectrum has a gap.