A note on monotonicity of mixed Ramsey numbers

TL;DR

This paper investigates the structure of mixed-Ramsey spectra, showing that the spectrum forms an interval under certain conditions related to the graphs G and H.

Contribution

It proves that the mixed-Ramsey spectrum is an interval when G is not a star and H lacks a pendent edge, clarifying a fundamental property of these colorings.

Findings

Spectrum is an interval if G is not a star.

Spectrum is an interval if H does not contain a pendent edge.

Provides conditions under which the spectrum's monotonicity holds.

Abstract

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 number of colors used by some $(G,H)$-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 pendent edge.