Sh:1182

• Golshani, M., Mirabi, M., & Shelah, S. From Shelah’s block - content to Hales - Jewett. Preprint. arXiv: 2104.05962
• 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.

• Version 2026-07-13_2 (11p)

@article{Sh:1182,
 author = {Golshani, Mohammad and Mirabi, Mostafa and Shelah, Saharon},
 title = {{From Shelah's block - content to Hales - Jewett}},
 note = {\href{https://arxiv.org/abs/2104.05962}{arXiv: 2104.05962}},
 arxiv_number = {2104.05962}
}