# Sh:1182

- Golshani, M., & Shelah, S.
*Iterated Ramsey bounds for the Hales-Jewett numbers*. Preprint. arXiv: 1912.08643 -
Abstract:

Consider the Hales-Jewett theorem. The k-dimensional version of it tells us that the combinatorial space \mathcal{U}_{M, \Lambda} = \{ \eta \mid \eta: M \to \Lambda \} has, under suitable assumptions, monochromatic k-dimensional subspaces, where by a k-dimensional subspace we mean there exist a partition \langle N_0, N_1, \cdots, N_k \rangle of M such that N_1, \cdots, N_k \neq \emptyset (but we allow N_0 to be empty) and some \rho_0: N_0 \to \Lambda, such that the subspace consists of those \rho \in \mathcal{U}_{M, \Lambda} such that for 0<l<k+1, \rho \restriction N_l is constant and \rho \restriction N_0= \rho_0.It seems natural to think it is better to have each N_{l}, 0<l<k+1 a singleton. However it is then impossible to always find monochromatic k-dimensional subspaces (for example color \eta by 0 if |\eta^{-1}\{\alpha \}| is an even number and by 1 otherwise). But modulo restricting the sign of each |\eta^{-1}\{\alpha \}|, we prove the parallel theoremâ€“ whose proof is not related to the Hales-Jewett theorem. We then connect the two numbers by showing that the Hales-Jewett numbers are not too much above the present ones. This gives an alternative proof of the Hales-Jewett theorem.

- Version 2021-01-02_2 (12p)

Bib entry

@article{Sh:1182, author = {Golshani, Mohammad and Shelah, Saharon}, title = {{Iterated Ramsey bounds for the Hales-Jewett numbers}}, note = {\href{https://arxiv.org/abs/1912.08643}{arXiv: 1912.08643}}, arxiv_number = {1912.08643} }