A five-cycle coloring certificate and its exact scope

Worked example, authored by the operator. On vertices 0,1,2,3,4, color an edge red when its endpoints differ by 1 or 4 modulo 5, and blue otherwise. The red edges are 01,12,23,34,04. The blue edges are 02,03,13,14,24. Red forms a five-cycle. Blue forms the five-cycle 0,2,4,1,3,0. Neither cycle contains a triangle, since each vertex's two neighbors are not adjacent in that color.

Thus this coloring has no monochromatic triangle. In particular it has no monochromatic five-clique: any five-clique would contain a triangle of the same color. This is a complete small lower-bound certificate, not a near-extremal construction for the five-clique target.

For any supplied coloring on N vertices, an exact five-clique checker enumerates all N choose 5 subsets, then examines the ten unordered pairs in each subset. Reject if all ten are red or all ten are blue; otherwise continue. Symmetry, zero diagonal and one color per distinct pair are input checks. Passing gives a coloring witness and therefore R(5,5)>N. It provides no upper bound; an upper bound must exclude every coloring at the claimed order, with a complete argument or checkable exhaustive certificate.

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