Spectra Explorer

This page studies edge-colorings of the complete graph K_n, in which every pair of the n vertices (dots) is joined by an edge. A mixed Ramsey problem fixes two forbidden subgraphs, F and H, and asks for edge-colorings with no monochromatic copy of F (every edge the same color) and no rainbow copy of H (every edge a different color). Too few colors force the first pattern; too many force the second. The mixed Ramsey spectrum MRS(K_n; F, H) is the set of all numbers of colors k for which some edge-coloring of K_n with exactly k colors avoids both, the object at the center of much of my research.

To explore the spectra

1. Pick a rule (set of forbidden subgraphs).
2. Select the number of dots (vertices) for your graph.
3. Choose a number of colors for the edges of your graph.
4. If that number combination works, an example graph edge-colored with exactly that many colors appears in the section below.

How these values were found

Each spectrum comes from an exhaustive computer search over every coloring of the edges, up to renaming the colors, and was cross-checked by a second, independently written program. For five or fewer vertices, that program lists every possible coloring outright. The computed spectra were also checked against the results in my papers. Every example coloring on this page is re-checked in your browser against each triangle or cycle as it is drawn. The search program, the independent checkers, and the computed results are available to download: spectra-explorer-code.zip.