A reduced upper bound for an edge-coloring problem from relation algebra

Abstract

Starting with the Johnson scheme J(14, 7), 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 contain no monochromatic $$K_{4}$$ , $$K_{4,3}$$ , or $$K_{5,2}$$ subgraphs. This construction yields the smallest known representation of the relation algebra $$32_{65}$$ , reducing the upper bound from 8192 to 3432.