Nineteen exact reflective and dihedral Ramsey numbers from Damnjanovic-Dordevic's tables

Sixteen previously unknown exact values, plus three that confirm the sibling theorem entries' predictions computationally, across five ordered-pattern families ( $P^{alt}$ , $S^{sc}$ , $C^{mon}$ , $M^{nest}$ , $K$ ) under dihedral and reflective group actions - each closing one open cell of Damnjanovic-Dordevic (arXiv:2607.06817, Tables 3-13). Five sit in cells the paper left without a conjecture. Full per-value table with regeneration commands, certificate hashes and referee verdicts in the evidence repo.

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Computation
Field: Permutational Ramsey theory
Posed by: Damnjanovic-Dordevic (open cells of Tables 3-13)
Year posed: 2026
Years open: 0y
Solved: 2026-08-13
Model: Claude Fable 5
Vendor: Anthropic
Collaborators: —
Verification: Site-confirmed
Publication: Announced
Significance: 5 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed $n$ , witnessed satisfiable at $n-1$ . They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the $K$ column. The three overlap cells are $R_{dih}(P_4^{alt},K_6)=16$ , $R_{dih}(P_3^{alt},K_9)=17$ and $R_{dih}(P_9^{alt},K_3)=17$ , each an instance of a sibling theorem; the remaining sixteen stand on their own certificates. Four further cells passed the producing solver but await their final referee leg and are not claimed. Open: every other cell of DD26's tables, all cyclic-action and online-Ramsey cells.

What the AI did

AI agents built the census of open cells from the paper's tables, encoded every instance using the authors' own vendored generator, orchestrated kissat solving with drat-trim verification, and refereed all nineteen values through six independent agent reviews — each referee wrote its own encoder from the definitions, zero shared code with the producing leg or each other. Human direction was limited to run design, operational supervision, and posting.

Verification

Half-reproduced by this site on 14 August 2026, referee code, witnesses and CNFs deliberately unread - and this note is precise about which half. All 19 instances regenerate from the vendored DD26 generator and match their pinned SHA-256 byte for byte. All 19 LOWER bounds were established independently: the five pattern families, both group actions and the monotone-embedding notion were implemented here from the DD26 paper's definitions alone, a witness coloring at $n-1$ was found for every value, and each witness was re-verified by a brute-force embedding search sharing no code with the encoder. The UNSAT side is where this site's reach ended: the instances are genuinely hard (the submitter's smallest took kissat nine minutes; this site's solver decided none within a 50-minute-per-instance budget), so the upper bounds rest on the bundle's own certificates - whose 19 drat-trim logs all read "s VERIFIED" with parsed dimensions matching the regenerated CNFs exactly - except for the three K-family cells, whose values are implied by the two sibling theorem entries this site verified in depth on 13 August. Every cell with a DD26 conjecture matches that conjecture's formula, checked by hand. Not covered: no human peer review - produced and refereed by AI agents in one pipeline (six referee legs, each with its own encoder); the Lean file is a statement anchor with zero proofs, by design. Four further cells await their final referee leg and are, correctly, not claimed.

Sources

Related entries

Related $R_{dih}(P_a^{alt},K_b)=1+(a-1)(b-1)$ , $age4$
Related $R_{dih}(P_3^{alt},K_b)=2b-1$

Submitted by ZestyWombat854 on 14 Aug 2026