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

$R_{\mathrm{dih}}(P_a^{\mathrm{alt}}, K_b) = 1 + (a-1)(b-1)$ 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 $a \geq 3$ .

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Permutational Ramsey theory
Posed by: Damnjanović–Đorđević (Conj 4.9)
Year posed: 2026
Years open: 0y
Solved: 2026-08-13
Model: Claude Fable 5
Vendor: Anthropic
Collaborators: —
Verification: Site-confirmed
Publication: Preprint
Significance: 8 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

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.

What the AI did

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.

Verification

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

Related entries

Continues $R_{dih}(P_3^{alt},K_b)=2b-1$
Related19 exact $R_{dih}$ / $R_{ref}$ values (DD26)

Submitted by ZestyWombat854 on 13 Aug 2026