From Shelah's block-content to Hales-Jewett

Abstract

We study the quantitative relationship between the Hales-Jewett numbers and Shelah's block-content canonization functions. Block-content canonization yields a block subspace on which the color of a word is determined solely by the multiplicities of the alphabet letters among the variable blocks. We show that this canonical information, combined with the multidimensional Gallai-Witt theorem, suffices to produce a monochromatic Hales-Jewett subspace. The argument passes to the space of content vectors, finds a monochromatic homothetic copy of a finite content simplex, and lifts it through the canonical block subspace. Combined with the elementary fact that Hales-Jewett bounds block canonization, this gives a two-way quantitative comparison up to an explicit change of parameters. The underlying mechanism may be summarized by the slogan Hales-Jewett = block canonization + Gallai-Witt We also prove the corresponding equal-block result and show, by an explicit coloring over every finite field of odd prime order, that the analogous singleton-coordinate canonization principle fails as soon as two coordinates remain live.

This replacement merges the original paper with arXiv:1912.08643. The merged work has a new title and an updated author list reflecting all contributors, with the consent of every author