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

The Hales-Jewett number HJ(t,r) is the least dimension n such that every r-coloring of the grid [t]^n contains a monochromatic combinatorial line.

We prove HJ(3,3) >= 22 and HJ(4,2) >= 14, improving the previous records 14 and 12.

The engine is an exact reduction: a coloring of [t]^n invariant under coordinate permutations descends to the discrete simplex of letter-count vectors, where a combinatorial line is precisely a corner tuple; the 4,387,586,157,901 lines of [3]^21 thereby compress to 1771 local conditions on 253 cells.

We prove that this symmetric class coincides with the class of one-weight colorings, those reading an integer-weighted count of the letters: a radix weight realizes every symmetric coloring, so the symmetric lower-bound problem is a one-dimensional homothety-avoidance problem, the case d=1 of Gallai's theorem.

This yields the closed-form bound HJ(t,r) >= ceil((G_r(S)-1)/D_S) in terms of the Gallai homothety numbers G_r(S), together with the new values G_3({0,1,3})=42, G_3({0,1,4})=57, G_2({0,2,3,5})=67, G_2({0,1,5,6})=80, and G_3({0,2,5})=77, giving HJ(3,3) >= 16 from a one-line certificate.

Further results: periodic one-weight palettes give HJ^[12](3,3) = HJ^[12](4,2) = infinity for lines with at most twelve active coordinates; the interval number HJ^(1)(3) is exactly 5; a Rado reading gives G_4({0,1,3}) >= 94 and R_4(z+2x=3y) >= 59; and a rainbow companion gives the anti-Hales-Jewett bound ah(3,4) >= 25.

A SAT program written for this article pushes the computation further: eighteen exact two-color Gallai numbers of four-point sets, up to G_2({0,1,6,7}) = G_2({0,3,4,7}) = 79; the exact three-color value G_3({0,1,5}) = 70; and the exact Rado numbers R_r(z+kx=(k+1)y) for 2 <= k <= 5 and r in {2,3}.

Every displayed certificate is verified by direct enumeration; certificates and verification scripts are available at this https URL.