Evaluating an avoidance number: A₃(4)=9, A₃(5) ≥ 18, and two brackets for oriented Ramsey numbers
Abstract
Duman, Gönül, Kaya, Saxena and Tamer introduce two finite Ramsey-type quantities as the inputs to a constant in their work on closed Ramsey numbers of small countable ordinals: the directed Ramsey number R(Kst(n),L₃) and an avoidance number A₃(k), the least m such that every 3-colouring of the arcs of Kst(m) admits a k-set S in which every vertex misses a colour on its arcs from S. They evaluate neither. We first observe that R(Kst(m),L₃) is the oriented Ramsey number r(Iₘ,L₃), for which exact values up to m=5 and the bound r(Iₘ,L₃) ≤ m²-m+3 are in print but are not cited in that paper. We then determine A₃(4)=9, the first value of the avoidance number beyond the trivial A₃(3)=3, against a published bracket whose upper end is the unknown three-colour Ramsey number R(4,4,4). The proof rests on a symmetry of A₃ that acts independently at each vertex; we also give the best upper bound we can prove with no search at all, A₃(4) ≤ 17, from a new obstruction on pairs of vertices, and the first nontrivial lower bound at k=5, A₃(5) ≥ 18. For the directed function we prove two brackets for which we could find no published counterpart, 29 ≤ r(I₆,L₃) ≤ 33 and 21 ≤ r(I₃,L₄) ≤ 25; in each the upper end is Ihringer, Rajendraprasad and Weinert's published bound and the lower end is new, the second being the cell they name as the natural next question and where only r(I₃,L₄) ≥ 9 was available. Every statement below has been machine-checked.
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Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
78799e3d1ff9f23cc04b86cd4c635c6d23cafbead94a03e1f36d9a76a1a125d5
Claim ledger
Stated results
AN1routine2026-08-22
The two definitions, transcribed from the source, with the reading pinned down by controls
AN2known2026-08-22
R(Kₙ^*, L₃) <= n² at every n, re-derived from the definition
AN3routine2026-08-22
A₃(3) = 3, against the published bracket [0.75, 17]
AN4known data2026-08-22
R(K₂^*,L₃) = 4 and R(K₃^*,L₃) = 9, each proved twice
AN5known data2026-08-22
Headline: R(K₄^*, L₃) = 15, so Larson-Mitchell's n² is not attained at n = 4
AN6routine2026-08-22
The relabelling and the extremal structure at m = n²-1, which is what makes the m = 15 refutation feasible
AN7candidate2026-08-23
A₃(4) >= 9, the one value here not found in print
AN8routine2026-08-22
Negative controls in both directions, plus definitional and perturbation controls
AN-sharpknown2026-08-22
r(Iₘ, L₃) <= m² - m + 3 for every m >= 3 (IRW's sharp bound), kernel-clean
AN-r5known data2026-08-22
r(I₅, L₃) = 23 – first formalization, no solver call
AN-r4-argroutine2026-08-22
r(I₄, L₃) = 15 re-proved from the argument alone (kernel-clean); routesₐgree_four ties it to the LRAT route
AN-r6candidate2026-08-23
29 <= r(I₆, L₃) <= 33
AN-l4-boundknown2026-08-22
The L₄ column: N⁺(v), N⁻(v) are Iₘ,L₃-free, I(v) is Iₘ₋₁,L₄-free, and the recursion cap4 m = 1 + 2·cap3 m + cap4 (m−1) they give
AN-l4-twoknown2026-08-22
r(I₂, L₄) = 8, both halves kernel-clean; the witness is the Paley tournament on ZZ₇
AN-l4-threecandidate2026-08-23
21 ≤ r(I₃, L₄) ≤ 25 — the cell IRW's Coda names as the next open problem; the lower end is a 20-vertex circulant and moves the published [9, 25]
AN-l4-tableknown2026-08-22
r(I₄,L₄) ≤ 54, r(I₅,L₄) ≤ 99, r(I₆,L₄) ≤ 164, and the check that the recursion is IRW's own v(m, 4)
AN-l4-controlsroutine2026-08-22
Negative controls: L₄ is the transitive tournament and not a path, the L₄ column is not the L₃ column, both inputs to the recursion are load-bearing, Oriented is not vacuous
AN-a34-uppercandidate2026-08-23
A₃(4) <= 28 by the counting argument (bracket now [9, 28], was [9, 62])
AN-a34-reductionroutine2026-08-23
Two SAT refutations would give A₃(4) = 9 (symmetry reduction + bridge, kernel-clean; budget 1.5 GB LRAT)
AN-a34-ninecandidate2026-08-29
A₃(4) = 9 — the first exact A₃ value beyond A₃(3) = 3, via the per-vertex colour WLOG
AN-a34-paircandidate2026-08-29
The pair obstruction |D(u,v)| ≤ 4 (tight) and A₃(4) ≤ 17 with no solver
AN-a35-gecandidate2026-08-29
A₃(5) ≥ 18 (native) and ≥ 15 (kernel-clean), and A₃(k) ≤ A₃(k+1)
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Two Ramsey-type functions defined, used once, and never evaluated in arXiv:2604.23433 (Duman, Gönül, Kaya, Saxena, Tamer, *On closed Ramsey numbers of small countable ordinals*, 25 Apr 2026), §"Coloring complete directed graphs". They are the two finite inputs to that paper's upper-bound constant M(n) = R(A₃(R(Kₙ^*, L₃)), 3).
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7