A note on the monotonicity of mixed Ramsey numbers

For two graphs, G and H, an edge coloring of a complete graph is (G,H)-good if there is no monochromatic subgraph isomorphic to G and no rainbow subgraph isomorphic to H in this coloring. The set of numbers of colors used by (G,H)-good colorings of K\"n 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 G is not a star and H does not contain a pendant edge.