Hales-Jewett theorem ($HJ(\alpha,k,n)$) states that for every coloring $f:\alpha^N\rightarrow k$ where $N$ is sufficiently large, there is an $n$-dimensional combinatorial subspace of $\alpha^N$ that is monochromatic for $f$. Where a combinatorial subspace of $\alpha^N$ is a set generated by a variable-word $v$ of length $N$ with $n$ variables. A variable-word is a sequence of $\{0,\cdots,\alpha\}$ (word) and $\{x_0,x_1,\cdots\}$ (variable); and the set generated by $v$ is the set of words obtained by substituting variables in $v$ by words.
Q1: Is there a generalization of HJ theorem saying that the combinatorial subspace can be chosen to satisfy certain property.
For example, it can be shown that: for every infinite sequence $n_0n_1\cdots$ with $n_0=2$, there is an $r\in\omega$, such that for every coloring $f:2^{n_0+\cdots+n_r}\rightarrow 2$, there is a $s\leq r$, a variable-word $v$ with $n_s$ many variables such that the combinatorial space generated by $v$ is monochromatic for $f$; moreover, the set of the smallest coordinates of each dimension of the combinatorial space (generated by $v$), namely $\{\min\{t: v(t) = x_m\}: m\in\omega\}$, equals $ \{n_0+\cdots+n_{s-1},\cdots, n_0+\cdots+n_s-1\}$. This generalizes $HJ(2,2,2)$. (A proof is too long to describe here and I wonder if this can be proved when $n_0>2$.)
Q2: Is there any application of such generalizations?
