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 $\langle N_0, N_1, \cdots, N_k \rangle$ 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 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.