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arXiv · 2104.05962

From Shelah's block-content to Hales-Jewett

Mohammad Golshani·Mostafa Mirabi·Saharon Shelah
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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.

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Mohammad Golshani, Mostafa Mirabi, Saharon Shelah. 2021-04-13. From Shelah's block-content to Hales-Jewett. https://arxiv.org/abs/2104.05962

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