Equivalence of the Multidimensional van der Waerden Theorem

Question

In 'Elemental Methods in Ergodic Ramsey Theory', exercise 1.12, it's asked to show that MvdW4 implies MvdW3, those being the assertions:
MvdW3: Let $k \in \mathbb{N}$. For any finite partition of $\mathbb{Z}^k$ one of the cells contains affine images of every finite set. An affine image here being a set of the form $a +bF$ where $a \in \mathbb{Z}^k$ and $b \in \mathbb{Z}$.
MvdW4: Let $k \in \mathbb{N}$ and $ \epsilon > 0$. If $X$ is a compact metric space and $T_1,\dots,T_k$ are commuting homeomorphisms of $X$ then there exists $x \in X$ and $n \in \mathbb{Z}, \, n \neq 0$, such that $\rho(x,T_i^n(x)) < \epsilon, \, 1 \leq i \leq k$, $\rho$ being the distance function in $X$.
I can see that MvdW4 implies that for any finite partition of $\mathbb{Z}^k$ one of the cells contains affine images of the set $\{(0,\dots,0),(1,0,\dots,0),\dots,(0,\dots,0,1)\}$, but can't see how to get to MvdW3.
Thanks!

Answer 1

If $F = \{f_1, f_2, \dots, f_t\}$ where $f_1, \dots, f_t \in \mathbb Z^k$, then we can use a finite coloring of $\mathbb Z^k$ to define a finite coloring of $\mathbb Z^t$, by giving a point $(x_1, x_2, \dots, x_t) \in \mathbb Z^t$ the color of $x_1 f_1 + x_2 f_2 + \dots + x_t f_t$.

If you can argue that one of the colors in this coloring of $\mathbb Z^t$ contains an affine image of $\{(1,0,0,\dots,0), \dots, (0,0,0,\dots,1)\}$, this gives us an affine image of $F$ in the original coloring of $\mathbb Z^k$.

In detail, we have $a \in \mathbb Z^t$, $b \in \mathbb Z$ such that the following points all have the same color in our coloring of $\mathbb Z^t$:

• $(a_1 + b, a_2, \dots, a_t)$,
• $(a_1, a_2+b, \dots, a_t)$,
• $\dots$,
• $(a_1, a_2, \dots, a_t + b)$.

This means in the original coloring of $\mathbb Z^k$, the following points all have the same color:

• $(a_1 f_1 + a_2 f_2 + \dots + a_t f_t) + b f_1$,
• $(a_1 f_1 + a_2 f_2 + \dots + a_t f_t) + b f_2$,
• $\dots$,
• $(a_1 f_1 + a_2 f_2 + \dots + a_t f_t) + b f_t$.

This is an affine image of $F$.