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arxiv: 2607.02226 · v1 · pith:KGTQEHSCnew · submitted 2026-07-02 · 🧮 math.CO

One-Weight Colorings, the Symmetric Class, and Lower Bounds for Hales--Jewett Numbers

Younes Mouhib This is my paper

Pith reviewed 2026-07-03 10:39 UTC · model grok-4.3

classification 🧮 math.CO
keywords Hales-Jewett numberssymmetric coloringsone-weight coloringsGallai theoremhomothetic copieslower boundsradix weightsbracket numbers
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The pith

Every symmetric coloring of the Hales-Jewett cube equals a one-weight coloring given by a single radix-weighted count of the letters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a coloring of the Hales-Jewett cube stays unchanged under any permutation of the coordinates if and only if it depends only on an integer-weighted tally of how often each letter appears. This identification converts the search for symmetric lower bounds on Hales-Jewett numbers into the task of coloring points on a line so that no homothetic copy of a fixed t-point set is monochromatic, which is precisely the one-dimensional case of Gallai's theorem. The reduction produces explicit new bounds such as HJ(3,3) at least 22 and HJ(4,2) at least 14, together with closed-form expressions for certain auxiliary Gallai numbers. It further shows that the symmetric class is sparse, containing only 36 members among the 1644 line-free two-colorings of the 3-by-3 cube.

Core claim

We prove that the two classes coincide -- a radix weight realizes every symmetric coloring -- so the symmetric lower-bound problem for the Hales--Jewett numbers is exactly a one-dimensional coloring problem about homothetic copies of a t-point set, the case d=1 of Gallai's theorem. Optimizing the weight yields HJ(3,3)≥22 and HJ(4,2)≥14, the latter in closed form from the new Gallai homothety numbers G2({0,2,3,5})=67 and G2({0,1,5,6})=80.

What carries the argument

the radix weight that reads an integer-weighted count of the letters and realizes every symmetric coloring

If this is right

  • The symmetric lower-bound search reduces to optimizing weights in the one-dimensional Gallai setting.
  • New concrete lower bounds follow: HJ(3,3) at least 22, HJ(4,2) at least 14, and several new G_r values at small parameters.
  • For lines whose active coordinates are bounded by K the bracket numbers become infinite.
  • The interval-active case is settled for every r by combining known bounds, and the one-active-coordinate case yields HJ^{(1)}(3)=5.
  • The (4,2) palette is shown to be an extremal bracket object plus one boundary scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence may allow systematic computer search for optimal weights at larger t and r by restricting to one dimension.
  • If the Collapse or symmetric-extremality conjectures hold, many Ramsey numbers would stabilize at the values already obtained from low-weight palettes.
  • The thinness result suggests that exhaustive enumeration of symmetric colorings remains feasible even when full Hales-Jewett enumeration is not.

Load-bearing premise

Every symmetric coloring arises from some integer radix weight without exception.

What would settle it

An explicit symmetric coloring of some [t]^n that cannot be reproduced by any choice of integer weights on the letter counts.

read the original abstract

A coloring of the Hales--Jewett cube $[t]^n$ is symmetric if it is invariant under all coordinate permutations, and one-weight if it reads only an integer-weighted count of the letters. We prove that the two classes coincide -- a radix weight realizes every symmetric coloring -- so the symmetric lower-bound problem for the Hales--Jewett numbers is exactly a one-dimensional coloring problem about homothetic copies of a $t$-point set, the case $d=1$ of Gallai's theorem. Optimizing the weight yields $\mathrm{HJ}(3,3)\ge22$ and $\mathrm{HJ}(4,2)\ge14$, the latter in closed form from the new Gallai homothety numbers $G_2(\{0,2,3,5\})=67$ and $G_2(\{0,1,5,6\})=80$; new values at three colors -- $G_3(\{0,1,3\})=42$, $G_3(\{0,1,4\})=57$ and $G_3(\{0,2,5\})\ge77$ -- give $\mathrm{HJ}(3,3)\ge16$ from a one-line certificate. An anatomy of the $(4,2)$ palette locates the source of its compression: it is an extremal object of the bracket regime plus a single boundary scale. An exhaustive census shows how thin the class is: of the $1644$ line-free $2$-colorings of $[3]^3$, exactly $36$ are symmetric. For lines with at most $K$ active coordinates the same machinery gives infinite bracket numbers, $\mathrm{HJ}^{[12]}(3,3)=\mathrm{HJ}^{[12]}(4,2)=\infty$, strictly beyond the sum-type ceilings $\kappa_{\mathrm{sum}}(3,3)=11$ and $\kappa_{\mathrm{sum}}(4,2)=10$; for lines whose active set is an interval the machinery is provably blind, the interval ceiling $\lambda(3,r)$ is settled for every $r$ by assembling the known bounds, and a SAT computation gives the exact value $\mathrm{HJ}^{(1)}(3)=5>4=\mathrm{HJ}(3)$. We close with the Collapse, diagonal-only, and symmetric-extremality conjectures and with open problems on optimal weights. Every certificate displayed in this note has been re-verified by direct enumeration, independently of any solver.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the classes of symmetric colorings and one-weight colorings of the Hales-Jewett cube [t]^n coincide, with every symmetric coloring realized by a radix weight. This equivalence reduces the symmetric lower-bound problem for Hales-Jewett numbers to the d=1 case of Gallai's theorem on homothetic copies of a t-point set. The paper derives new lower bounds including HJ(3,3)≥22 and HJ(4,2)≥14 (the latter from explicit G_2 values), supplies new G_3 values, gives an exhaustive census of the 36 symmetric line-free 2-colorings of [3]^3, establishes infinite bracket numbers HJ^[12](3,3)=HJ^[12](4,2)=∞, settles interval ceilings λ(3,r), and verifies all displayed certificates by direct enumeration independent of solvers. It closes with three conjectures and open problems on optimal weights.

Significance. If the equivalence holds, the result supplies a structural simplification that converts the symmetric HJ problem into a one-dimensional optimization over weights, directly yielding improved explicit lower bounds and infinite families for restricted active-coordinate sets. The independent direct-enumeration verification of all certificates and the [3]^3 census strengthens the computational claims. The explicit reduction to Gallai homotheties and the identification of extremal palettes (e.g., the (4,2) case) provide concrete new tools and conjectures for further work in Ramsey theory.

minor comments (3)
  1. [Introduction] §1 (Introduction): the Gallai homothety numbers G_d(S) are invoked in the abstract and early bounds without a self-contained definition or forward reference; a one-sentence recall of the definition would improve accessibility.
  2. The three conjectures (Collapse, diagonal-only, symmetric-extremality) are named in the final paragraph but not restated; a brief one-sentence formulation of each would clarify the open problems without lengthening the manuscript.
  3. Table or census section: the claim that exactly 36 of the 1644 line-free 2-colorings of [3]^3 are symmetric is central to the thinness statement; confirming that the enumeration code or method is described (even briefly) would allow independent reproduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and accurate summary of the manuscript, the positive significance assessment, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit proof

full rationale

The paper supplies a direct proof that symmetric colorings of [t]^n coincide exactly with one-weight colorings realized by radix weights, thereby reducing the symmetric HJ lower-bound problem to the d=1 Gallai case on homothetic copies. This equivalence is established rather than presupposed. Lower bounds such as HJ(3,3)≥22 and HJ(4,2)≥14 are obtained by optimizing the proven weight construction and are supported by exhaustive census, direct enumeration re-verification of all certificates, and independent SAT computations. No load-bearing self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard statement of Gallai's theorem in dimension 1 and introduces the radix-weight realization as its main new step; the weights themselves are free parameters chosen by optimization to produce the reported bounds.

free parameters (1)
  • radix weight = optimized per instance (e.g., for {0,2,3,5} and {0,1,5,6})
    Integer weights on the t symbols are optimized to produce the largest lower bound obtainable from the resulting one-weight coloring.
axioms (1)
  • domain assumption Gallai's theorem holds in dimension d=1
    The abstract states that the symmetric lower-bound problem reduces exactly to the d=1 case of Gallai's theorem.

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