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

Abstract: A coloring of the Hales--Jewett cube $[t]<sup>n$ 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 $[3]<sup>3$, exactly $36$ are symmetric. For lines with at most $K$ active coordinates the same machinery gives infinite bracket numbers, $$\mathrm{HJ}<sup>{[12]}(3,3)=\mathrm{HJ}<sup>{[12]}(4,2)=\infty$$, strictly beyond the sum-type ceilings $κ<em>{\mathrm{sum}}(3,3)=11$ and $κ</em>{\mathrm{sum}}(4,2)=10$; for lines whose active set is an interval the machinery is provably blind, the interval ceiling $λ(3,r)$ is settled for every $r$ by assembling the known bounds, and a SAT computation gives the exact value $\mathrm{HJ}<sup>{(1)}(3)=5&gt;4=\mathrm{HJ}(3)$. 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.

• The paper reveals that every symmetric coloring is of one-weight type, collapsing the hierarchy and simplifying extremal Hales–Jewett problems.
• It employs combinatorial reductions, radix-weight representations, and SAT-solving to produce new lower bounds such as HJ(3,3)≥22 and HJ(4,2)≥14.
• The study connects symmetric colorings with one-dimensional Gallai problems, offering theoretical insights and practical strategies for tackling Ramsey-type challenges.

One-Weight Colorings and the Symmetric Class: Structural Insights and Hales–Jewett Lower Bounds

Overview and Main Results

The paper "One-Weight Colorings, the Symmetric Class, and Lower Bounds for Hales–Jewett Numbers" (2607.02226) advances the structural and quantitative analysis of colorings in the Hales–Jewett cube $[t]^n$, unifying and elucidating the roles of symmetric and one-weight colorings. The key result asserts that every symmetric coloring is of one-weight type, collapsing the apparent hierarchy between genuinely symmetric colorings and those defined via an integer-weighted sum of letter counts. This exactly characterizes the symmetric coloring problem for Hales–Jewett numbers as a one-dimensional coloring problem related to Gallai's theorem.

By optimizing over one-weight (equivalently, symmetric) colorings, the paper obtains strong lower bounds for Hales–Jewett numbers---notably $HJ(3,3)\geq22$ and $HJ(4,2)\geq14$---that surpass previous records and the classic van der Waerden (vdW) reductions. Additionally, the work structurally analyzes the space of symmetric line-free colorings and elucidates compression mechanisms responsible for these new lower bounds.

The methodology encompasses combinatorial reductions, categorical structure of symmetric actions, computational verification (including SAT-solving), and mapping to classic theorems on homothetic copies in integers.

Symmetric and One-Weight Colorings: A Collapsed Hierarchy

A central contribution is the full identification of symmetric colorings with one-weight colorings. Formally, a coloring of $[t]^n$ is symmetric if it is invariant under all coordinate permutations, and is one-weight if it depends only on a linear combination (with integer coefficients) of letter frequencies in the word, post-composed with a coloring on $\mathbb{Z}$. By constructing a radix-weight representation with sufficiently large coefficients (depending polynomially on $n$), every symmetric coloring can be encoded via a one-weight map.

This equivalence is formalized in Theorem 3.5, which also demonstrates that, up to scaling and translation, there is no loss of generality in restricting attention to one-weight colorings for the extremal threshold problem. The same theorem clarifies that multi-weight constructions don't escape the one-weight/symmetric class, as functions of multiple such weightings also factor through the type simplex. The essential implication is that the previously conjectured infinite hierarchy by rank collapses after the one-weight layer.

Quantitatively, symmetric colorings are shown to be extremely scarce: for $(t,r)=(3,2)$ and $n=3$, only 36 out of 1644 line-free 2-colorings are symmetric, emphasizing the exceptional nature of symmetric extremal colorings.

The Gallai Reduction and New Lower Bounds

By exploiting the one-weight/symmetric structure, the lower bound question for Hales–Jewett numbers is reduced to a one-dimensional Gallai theorem: avoiding monochromatic homothetic copies of a fixed $t$-point subset $S$ under colorings of integer intervals. This connection enables derivation of closed-form lower bounds using Gallai homothety numbers $HJ(3,3)\geq22$0, and the results are sharp when the arithmetic progression structure is optimal, but can be substantially improved using non-arithmetic sets and weights.

Highlights of numerical improvements:

• $HJ(3,3)\geq22$1: A weight $HJ(3,3)\geq22$2 with an explicit 3-color palette yields a line-free coloring in $HJ(3,3)\geq22$3, improving beyond the standard van der Waerden bound ($HJ(3,3)\geq22$4) and previously published record ($HJ(3,3)\geq22$5).
• $HJ(3,3)\geq22$6: With new Gallai numbers $HJ(3,3)\geq22$7 and $HJ(3,3)\geq22$8, the bound exceeds the arithmetic projection ($HJ(3,3)\geq22$9).
• $HJ(4,2)\geq14$0: Established in closed form via $HJ(4,2)\geq14$1, indicating possible further gains.
• Systematic computational verification confirms the absence of monochromatic homothets/corner tuples in these constructions.

A structural "compression" effect is identified in the $HJ(4,2)\geq14$2 case: the extremal palette is explained as a "bracket regime" object able to reach beyond periodic block limitations by a single scale, producing a concise periodic certificate that demonstrates the bound. This is not paralleled in $HJ(4,2)\geq14$3, where a larger gap remains between periodic constructions and the full symmetric reach.

Bracket and Interval Variants: Expanded Landscape

The paper systematically studies restricted variants of the problem, notably:

• Bracket Hales–Jewett numbers $HJ(4,2)\geq14$4, counting only lines with up to $HJ(4,2)\geq14$5 active coordinates.
• Interval Hales–Jewett numbers $HJ(4,2)\geq14$6, restricting to lines whose active sets are unions of at most $HJ(4,2)\geq14$7 intervals.

Key findings include:

• For one-weight/symmetric colorings, periodic block palettes can be constructed to avoid lines with up to $HJ(4,2)\geq14$8 active coordinates for significant $HJ(4,2)\geq14$9 (e.g., $[t]^n$0).
• For interval lines, symmetric machinery is "blind": $[t]^n$1, so all nontrivial interval avoidance must be accomplished by asymmetric colorings.
• The interval ceiling for $[t]^n$2 exhibits parity phenomena unexplained by traditional approaches: $[t]^n$3 for $[t]^n$4 odd, $[t]^n$5 for $[t]^n$6 even.

Implications and Theoretical Insights

The identification of one-weight and symmetric colorings has several implications:

• Reduction in dimensionality: Extremal coloring problems for the Hales–Jewett cube reduce from exponential to polynomial (simplex-sized) domains.
• Combinatorial structural unification: The classification of symmetric colorings as "weight–periodic" functions aligns the monochromatic problem for high-dimensional cubes with classical Ramsey problems in one dimension.
• Genuine separation of bounds: New constructions using non-arithmetic weights surpass the best known vdW-based lower bounds, firmly establishing that "arithmetic progressions" are suboptimal for certain parameter regimes.
• Guide for computational search: The collapsed symmetry/weight hierarchy dramatically narrows the landscape for computational search for extremal colorings.

From an algorithmic perspective, the techniques set a foundation for efficiently searching for extremal colorings within the highly compressed simplex domain, as opposed to the exponentially sized Hales–Jewett cube.

Open Questions and Future Directions

The paper concludes with detailed conjectures and open questions:

• Collapse conjecture: Does $[t]^n$7 universally?
• Symmetric extremality conjecture: Is the symmetric threshold always tight, i.e., does $[t]^n$8 for all $[t]^n$9?
• Diagonal-only conjecture: Can the only monochromatic line in some coloring of $\mathbb{Z}$0 up to $\mathbb{Z}$1 be the diagonal?
• Characterization of optimal weights and palettes: Which sets $\mathbb{Z}$2 maximize the Gallai ratio? Are there categorical structures (e.g., orbits of affine maps) determining the most compressible sets?
• Quantitative gaps: What is the maximal bracket ceiling achievable for given parameters, and can non-periodic or quasi-periodic palettes exceed the observed block limits?
• Extension of compressibility and symmetry structure to hypergraph density settings and other Ramsey-type problems.

These questions are nontrivial: solving them will require a combination of advanced combinatorial enumeration, computational proof techniques (SAT, MaxSAT, symmetry breaking), and potentially analytic number-theoretic tools to bridge lower-bound gaps.

Conclusion

This work delivers a significant consolidation of symmetric and one-weight colorings, reshaping the approach to lower bounds for Hales–Jewett numbers. The technical results directly reframe the extremal theory for the cube in terms of classic unidimensional combinatorial theorems, yielding explicit, verifiable, and in several cases optimal lower bounds for central Ramsey-type numbers. This unification provides both deeper theoretical understanding and practical tools for future advances in Ramsey theory, coloring complexity, and combinatorial geometry. The scarcity and structure of symmetric colorings highlighted in the census points to rich algebraic and geometric phenomena yet to be fully exploited.

The broader implication is the prospect that, even in highly symmetric high-dimensional combinatorial problems, optimal or near-optimal solutions may lie in extremely sparse, structurally constrained families—suggesting promising directions for future research in both extremal combinatorics and algorithmic search.