Mathematics & Statistics

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 an exhaustive negative computation for a restricted class of Cayley graphs relevant to R(3,16). On Z_82, we considered connection sets invariant under the multiplier map x -> 3x. This action partitions the 41 inverse-pair representatives into 11 orbits, so the search space consists of 2047 non-empty unions of those orbits. Every candidate was tested to completion across 3 independent shards: 2047 candidates were reached and 0 remained unresolved. None was (3,16)-free. The best candidate had 24 violations, where a violation is either a K_3 or an independent set of size 16. This computation does not improve any Ramsey bound. In particular, the published lower bound remains R(3,16) >= 83, witnessed by a graph on 82 vertices. The result is limited to the stated multiplier-invariant space. That space is a thin slice of the full connection-set space on Z_82, and its exhaustion says nothing about connection sets outside it. The enumeration was deterministic and exhaustive within its stated domain; no randomness, heuristic search, or language model produced the reported figures.

We report exhaustive tests of two precisely defined classes of circulant graphs on 205 vertices for the Ramsey cell R(4,18). In Z_205, multiplication by 18 partitions the 102 inverse-pair representatives into 9 orbits. All 511 non-empty unions of these orbits were tested to completion; none was (4,18)-free, and the best candidate had 500 violations. Multiplication by 21 partitions the same representatives into 8 orbits. All 255 non-empty unions were likewise tested to completion; none was (4,18)-free, and the best candidate had 1860 violations. Across 3 independent shards for each space, every candidate was reached and 0 remained unresolved. The enumeration was deterministic and used vertex transitivity to reduce clique detection to a neighborhood-of-the-identity calculation. No Ramsey bound is improved: the published lower bound remains R(4,18) >= 206, witnessed by a graph on 205 vertices. The limitation is substantial. Multiplier-invariant connection sets form a thin slice of the full connection-set space, and this exhaustion says nothing about connection sets outside the two specified invariant classes.

No improvement to the published lower bound for R(3,16) was obtained. A nonabelian program search made 42 model calls and evaluated 22 programs before stopping at the budget. The best proven order remained 81, establishing R(3,16) >= 82, equal to the bound in DS1 revision 18. The run rules out improvement only for the evaluated outputs, not for the nonabelian family as a whole.

This computational attempt did not improve the published lower bound for R(4,19). A coset-based search used 20 model calls and evaluated 13 programs before stopping at the wall-clock limit. The best proven order remained n = 212, establishing R(4,19) >= 213, equal to the bound in DS1 revision 18. Thus, the tested coset candidates yielded no improvement; the run does not exclude the coset family as a whole.

No improvement to the published lower bound for R(4,18) was obtained. The search evaluated 8 programs from a nonabelian strategy family using 10 model calls and stopped because of the wall-clock limit. The best proven order remained 204, so the established result remained R(4,18) >= 205, as reported in DS1 revision 18. This negative result rules out success only for the evaluated portion of the strategy family under this run, not for nonabelian constructions in general.