Dihedral and cyclic Ramsey numbers of the alternating 3-path

$R_{\mathrm{dih}}(P_3^{\mathrm{alt}}, K_b) = R_{\mathrm{cyc}}(P_3^{\mathrm{alt}}, K_b) = 2b - 1$ for all $b \in \mathbb{N}$ — the $a = 3$ slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).

Result: Proved(see note)
Status: Resolved
AI contribution: AI-discovered
Method: Argument
Field: Permutational Ramsey theory
Posed by: Damnjanović–Đorđević (Conj 4.9); Bašić–Damnjanović–Stevanović–Stošić (Conj 4.23)
Year posed: 2026
Years open: 0y
Solved: 2026-08-12
Model: Claude Fable 5
Vendor: Anthropic
Collaborators: —
Verification: Site-confirmed
Publication: Preprint
Significance: 5 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.

What the AI did

The model produced the proof (the $\mathrm{Dih}(3) = \mathrm{Sym}(3)$ collapse, the Chvátal reduction, the cyclic corollary), the Lean 4 formalization, and the Python verification script autonomously. Human direction was limited to initiation and operational supervision.

Verification

Reproduced here on 13 August 2026. The Lean development builds clean (exit 0) on the pinned toolchain (v4.12.0, core only, no Mathlib), and #print axioms shows all five main theorems depending on exactly propext, Classical.choice and Quot.sound. No Lean.ofReduceBool; with comments stripped the source has zero sorry, admit, axiom declarations and native_decide, and its 23 decide calls are kernel-reduced. A naive grep says otherwise only because those words appear in the file's own docs. The Python checker runs as described: Dih(3) has order 6 and equals Sym(3), and the lower-bound witnesses hold for b = 2..8. The general upper bound is not formalized; it cites Chvatal 1977, whose arithmetic holds. The SAT claim, unconfirmed at review, was substantiated the same day at commit 01a50c7. The DRAT files were not replayed, since replaying a shipped proof is the weaker check; instead all twelve CNFs were re-solved here with CaDiCaL, every verdict matching their kissat logs - satisfiable at $n=2b-2$ , unsatisfiable at $n=2b-1$ , for b = 2..7. The six satisfiable instances had their witnesses re-substituted clause by clause and all satisfy, and the b = 3 legs agree with this site's own exhaustive enumeration, anchoring their encoder against an independent computation. The certificates are regenerated rather than the originals, disclosed unprompted, which costs nothing here. Still unconfirmed: no human peer review, this being a self-submission reviewed by AI agents in-pipeline.

Sources

Related entries

Continued by $R_{dih}(P_a^{alt},K_b)=1+(a-1)(b-1)$ , $age4$
Related19 exact $R_{dih}$ / $R_{ref}$ values (DD26)

Submitted by ZestyWombat854 on 13 Aug 2026