{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "title": "Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)", "canonical_statement": "$R_{\\mathrm{dih}}(P_a^{\\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ for all $a \\geq 4$, $b \\geq 1$ — the $a \\geq 4$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the $a = 3$ case (see sibling entry), this resolves Conjecture 4.9 in full for $a \\geq 3$.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:event", "problem_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "type": "proved", "status": "resolved", "occurred_at": "2026-08-13T00:00:00.000Z", "title": "GitHub repo (proof + referee reports + verification code + partial Lean formalization)", "summary": "The dihedral case only, for every $a \\ge 4$ and $b \\ge 1$; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim $1+(a-1)(b-1)$ for all $a \\ge 3$; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue $R_{cyc}(P_a^{alt}, K_b)$ for $a \\ge 4$ remains open. The engine is a self-contained inequality of independent interest: for any graph on a linearly ordered vertex set, the alternating-path reach statistics satisfy $\\sum_m [P(m)+Q(m)] \\ge 2|E(G)|$, from which the theorem falls out by averaging and a pivot decomposition.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:attempt" ], "method_family_ids": [ "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:method" ], "source_assertion_ids": [ "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:assertion" ] } ], "attempts": [ { "id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:attempt", "problem_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "model": "Claude Fable 5", "provider": "Anthropic", "attempted_at": "2026-08-13T00:00:00.000Z", "human_collaborators": [], "exposure": "unknown", "prompt_public": null, "outcome": "The proof was produced by a sealed, multi-agent research process: independently-launched Claude agents across three rounds, convergent results cross-validated. Two independent AI referee agents reviewed it dual-blind; both CONFIRMED. Human direction was limited to run design, operational supervision, and manual re-derivation of two write-up fixes.", "failed_routes": [], "artifacts": [ { "label": "Damnjanovic and Djordjevic, Computation of small reflective and dihedral Ramsey numbers (Conjecture 4.9)", "url": "https://arxiv.org/abs/2607.06817", "kind": "problem-record", "license": null } ], "sources": [ { "label": "GitHub repo (proof + referee reports + verification code + partial Lean formalization)", "url": "https://github.com/ZestyWombat854/alternating-path-ramsey", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)", "url": "https://vibemathed.com/problem/dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:verification", "problem_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "solution_event_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "Reproduced by this site on 13 August 2026, working from the pinned statement alone - the proof's machinery, both referee reports and the shipped CNFs were not consulted by the checker. Confirmed independently: the orbit anchor ($|Dih(a)$-orbit of $P_a^{alt}| = a$ for a = 3..14); the Ramsey value at nine (a,b) cells in both directions - a good coloring exists at $n = (a-1)(b-1)$ and none at $n+1$ - exhaustively over every 2-coloring at (4,2), (5,2), (6,2), (7,2) and (4,3), and via an independently written CNF encoding solved with CaDiCaL at (8,2), (5,3), (6,3) and (4,4); and the proof's load-bearing inequality, the Aggregate Sum Theorem, by a third implementation built from the P/Q definitions rather than the recursion, over all 33,868 labeled graphs on up to six vertices - zero violations, minimum slack 0, so the bound is tight. The prose proof was also read here in full and every algebraic step traced. Not covered by the tier: the general argument has no human peer review - produced by a sealed multi-agent Claude run and refereed dual-blind by two AI agents in the same pipeline (both CONFIRMED; one non-fatal bug and one cosmetic slip found and repaired inline, originals kept). The Lean part is partial by its own declaration - four side lemmas, zero sorry or native_decide, standard axioms, source-audited here but not compiled (pinned v4.30.0 + Mathlib, no CI runs). The main theorems are not formalized; there, the referee reports and this site's checks are the verification.", "sources": [ { "label": "VibeMathed: Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)", "url": "https://vibemathed.com/problem/dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "GitHub repo (proof + referee reports + verification code + partial Lean formalization)", "url": "https://github.com/ZestyWombat854/alternating-path-ramsey", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "GitHub repo (proof + referee reports + verification code + partial Lean formalization)", "url": "https://github.com/ZestyWombat854/alternating-path-ramsey", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)", "url": "https://vibemathed.com/problem/dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1:signal:watch_only", "problem_id": "vibemathed:dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }