A Reduced Upper Bound for an Edge-coloring Problem from Relation Algebra

TL;DR

This paper presents a new, smaller upper bound for a specific edge-coloring problem related to relation algebra, achieved through a novel cyclic coloring construction that avoids certain monochromatic triangles.

Contribution

It introduces a new construction that reduces the upper bound for representing a particular relation algebra from 8192 to 3432.

Findings

Constructed an edge-coloring of K_N with N=3432 avoiding monochromatic blue triangles.

Established a cyclic coloring of K_17 with specific forbidden subgraphs.

Achieved the smallest known representation of relation algebra 32_65.

Abstract

We construct an edge-coloring of $K_{N}$ (for $N = 3432$) in colors red, dark blue, and light blue, such that there are no monochromatic blue triangles and such that the coloring satisfies a certain strong universal-existential property. The edge-coloring of $K_{N}$ depends on a cyclic coloring of $K_{17}$ whose two color classes are $K_{4}$-, $K_{4,3}$-, and $K_{5,2}$-free. This construction yields the smallest known representation of the relation algebra $32_{65}$, reducing the upper bound from 8192 to 3432.