Papers

We report an exhaustive computation in two restricted classes of circulant graphs on 111 vertices for the Ramsey cell R(3,20). No Ramsey bound is improved. The published lower bound R(3,20) >= 112 is witnessed by a graph on 111 vertices. We tested every non-empty multiplier-invariant connection set in the specified classes. For multiplication by 26 in Z_111, the 55 inverse-pair representatives split into 13 orbits, yielding 8191 non-empty unions; all 8191 candidates were completed, with no unresolved candidate, and none was (3,20)-free. The best candidate had 33 violations, where a violation is a K_3 or an independent set of size 20. For multiplication by 41, the representatives split into 7 orbits, yielding 127 non-empty unions. All 127 candidates were completed, again with no unresolved candidate, and none was (3,20)-free; the best had 72 violations. The computation was deterministic and used vertex transitivity to reduce clique detection to the identity neighbourhood. These exhaustive results apply only to the stated multiplier-invariant spaces, which are thin slices of the full connection-set space.

We report exhaustive negative searches for R(4,19) witnesses among two precisely defined classes of circulant graphs on 213 vertices. This work does not improve any Ramsey bound. The published lower bound remains R(4,19) ≥ 214, witnessed by a graph on 213 vertices. In Z_213, multiplication by 20 partitions the 106 inverse-pair representatives into 11 orbits. Every one of the 2047 non-empty unions of these orbits was tested to completion; none was (4,19)-free, and the best candidate had 140 violations. Multiplication by 11 gives 4 orbits and 15 non-empty unions. All 15 were tested to completion; none was (4,19)-free, and the best candidate had 54740 violations. A violation is a K_4 or an independent set of size 19. Each enumeration was deterministic, was distributed across 3 independent shards, and ended with 0 unresolved candidates. These conclusions apply only to the stated multiplier-invariant connection sets. Such sets form a thin slice of the full connection-set space, so the computation says nothing about circulant connection sets outside these classes or about general graphs on 213 vertices.