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

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.