Symmetric colorings of Hales-Jewett cubes coincide with one-weight colorings, reducing the symmetric lower-bound problem to 1D Gallai homothety coloring and yielding HJ(3,3)≥22 and HJ(4,2)≥14.
An upper bound for the Hales-Jewett number HJ(4,2)
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abstract
We show that for $n$ at least $10^{11}$, any 2-coloring of the $n$-dimensional grid $[4]^n$ contains a monochromatic combinatorial line. This is a special case of the Hales-Jewett Theorem, to which the best known general upper bound is due to Shelah; Shelah's recursion gives an upper bound between $2 \uparrow \uparrow 7$ and $2 \uparrow \uparrow 8$ for the case we consider, and no better value was previously known.
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math.CO 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Improved lower bounds HJ(3,3)≥22 and HJ(4,2)≥14 obtained from explicit symmetric colorings whose line-freeness reduces to a SAT check on letter-count vectors.
citing papers explorer
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One-Weight Colorings, the Symmetric Class, and Lower Bounds for Hales--Jewett Numbers
Symmetric colorings of Hales-Jewett cubes coincide with one-weight colorings, reducing the symmetric lower-bound problem to 1D Gallai homothety coloring and yielding HJ(3,3)≥22 and HJ(4,2)≥14.
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Improved Lower Bounds for the Hales-Jewett Numbers via Symmetric Colorings
Improved lower bounds HJ(3,3)≥22 and HJ(4,2)≥14 obtained from explicit symmetric colorings whose line-freeness reduces to a SAT check on letter-count vectors.