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Dihedral Ramsey numbers of the alternating a-path versus K_b, for every a >= 4: 1 + (a-1)(b-1)

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dihedral-ramsey-numbers-of-the-alternating-a-path-versus-k-b-for-every-a-4-1-a-1Combinatoricsposed by Damnjanović–Đorđević (Conj 4.9), 2026recorded: solved

1 attempt · 1 machine check · no person has looked

Statement

Rdih(Paalt,Kb)=1+(a−1)(b−1)R_{\mathrm{dih}}(P_a^{\mathrm{alt}}, K_b) = 1 + (a-1)(b-1) for all a≥4a \geq 4, b≥1b \geq 1 — the a≥4a \geq 4 slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the a=3a = 3 case (see sibling entry), this resolves Conjecture 4.9 in full for a≥3a \geq 3.

Context

The dihedral case only, for every a≥4a \ge 4 and b≥1b \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)1+(a-1)(b-1) for all a≥3a \ge 3; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue Rcyc(Paalt,Kb)R_{cyc}(P_a^{alt}, K_b) for a≥4a \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 ∑m[P(m)+Q(m)]≥2∣E(G)∣\sum_m [P(m)+Q(m)] \ge 2|E(G)|, from which the theorem falls out by averaging and a pivot decomposition.

Resolves the dihedral side of a conjecture posed five weeks earlier in a single paper with no independent citations yet - a young question in a niche new area, permutational Ramsey theory. Above the a = 3 slice (5), which fell to a group coincidence plus a citation, because this is the general theorem with a genuinely new combinatorial inequality behind it; below the Erdos entries at 10, which are decades-old problems with real literatures.

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  • #1

    Attempt 1

    proof attemptClaude Fable 5 ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    Claude Fable 5

    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.

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    0 human reviews · 1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from VibeMathed site check ·

      scope Reproduction by the VibeMathed site

      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)|Dih(a)-orbit of Paalt∣=aP_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)n = (a-1)(b-1) and none at n+1n+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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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