Iterated Ramsey bounds for the Hales-Jewett numbers

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 $⟨N_0,N_1,\cdots ,N_k⟩$ of $M$ such that $N_1,\cdots ,N_k\ne \varnothing$ (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 $0<l<k+1,\rho \upharpoonright N_l$ is constant and $\rho \upharpoonright N_0={\rho }_0.$ 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 theorem -- 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.

This paper has been withdrawn by the authors as it has been combined with another work and is now superseded by arXiv:2104.05962

References in corpus (2)

• On what I do not understand (and have something to say): Part I
• On what I do not understand (and have something to say), model theory