The Chromatic Spectrum of a Ramsey Mixed Hypergraph

We extend known structural theorems, primarily a result of Axenovich and Iverson, for the strict edge colorings of the complete graph $K_n$ which avoid monochromatic and rainbow triangles to discover recursive relationships between the chromatic spectra of the bihypergraphs modeling this coloring problem. In so doing, we begin a systematic study of coloring properties of mixed hypergraphs derived from coloring the edges of a complete graph $K_n$ in such a way that there are no rainbow copies of $K_r$ and no monochromatic copies of $K_m$, where $n\ge r \ge 3,$ $ n\ge m \ge 3.$ We present the chromatic spectra of the bihypergraph models of $K_n$ for $4\le n \le 12$ and $r=m=3$. This study fits in the larger context of investigating mixed hypergraph structures that realize given spectral values, as well as investigations of the sufficiency of the spectral coefficients in obtaining recursive relationships without the need to subdivide them further into terms that count finer distinctions in the feasible partitions of the hypergraph. The bihypergraphs arising in this simplest case where $r=m=3$ have spectra that are gap free and which do allow a recursive relationship, albeit a complicated one. The continuation of this project in future work will examine if both of these facts remain true for derived Ramsey Mixed Hypergraphs corresponding to larger $r$ and $m$.