How can I prove these two versions of Van der Waerden theorem are equivalent?

Question

1. general version of Van der Waerden theorem is If $ℕ=A1∪⋯∪Ak$ is finitely partitioned then at least one of the sets A1,…,Ak contains arbitrarily long arithmetic progression.
2. Another version is if A $\subseteq \mathbb{N}$, and A is AP-rich (having arbitrarily long arithmatic progression), then A is finitely partitioned, $A=C1\cup C2\cup C3\cdot\cdot\cdot \cup Cr$, then at least one of Ci contains arbitrarily long AP(arithmatic progression).

I know how to prove 2 $\Rightarrow$ 1, because 1 is just a special case of 2, but how can I derive 2 from 1 to finish my proof of their equivalence?

Answer 1

One way is first to prove that (1) implies the finitistic version:

(1a) Let $r,k\in\Bbb N$ be given. Then there is a $W(r,k)\in\Bbb N$ such that if $m\ge W(r,k)$, and $[m]$ is colored with $r$ colors, then there is a monochromatic AP of length $k$.

Then you can use this to prove a corresponding result for arbitrary AP-rich $A$ instead of $\Bbb N$, replacing $[m]$ with an arithmetic progression of length $m$ in $A$. The final step is to show that this in turn implies (2); this is pretty straightforward, and I’ll leave it to you.

The first step is the hardest. The proof that I know is topological. Suppose that (1a) fails for some $r$ and $k$. For $n\in\Bbb N$ endow $C_n=[r]$ with the discrete topology, and let $X=\prod_{n\in\Bbb N}C_n$ with the product topology. Each point of $X$ corresponds to a coloring of $\Bbb N$: $x=\langle x_n:n\in\Bbb N\rangle\in X$ assigns the color $x_n$ to $n$. Note that $X$ is the product of countably many compact metrizable spaces, so $X$ is compact and metrizable and therefore sequentially compact.

Since (1a) fails, there are arbitrarily large $m\in\Bbb N$ and $r$-colorings $c_m:[m]\to[r]$ of $[m]$ with no monochromatic AP. Let $\langle m_i:i\in\Bbb N\rangle$ be a strictly increasing sequence of such ‘bad’ $m$, and for each $i\in\Bbb N$ let $x^{(i)}\in X$ be such that $x^{(i)}\upharpoonright[m_i]=c_i$. $X$ is sequentially compact, so the sequence $\langle x^{(i)}:i\in\Bbb N\rangle$ has a subsequence converging to some $x\in X$.

By hypothesis the coloring $x$ has arbitrarily long monochromatic APs, so in particular it has a monochromatic AP of length $k$, say $M$. Then $x\upharpoonright M$ is constant, say at $p\in[r]$. Let

$$B=\{y\in X:y_n=p\text{ for each }n\in M\}\;;$$

$B$ is an open nbhd of $x$ in $X$, so there is an $\ell\in\Bbb N$ such that $x^{(i)}\in B$ whenever $i\ge\ell$. But then $M$ is a monochromatic AP of length $k$ for the coloring $c_\ell$ of $[m_\ell]$, contradicting the choice of $c_\ell$ and establishing (1a).

Answer 2

Consider the following definition:

The notation $[N]$ is shorthand for the initial set $\{1,2,\cdots, N\}$. A $r$-colouring of $[N]$ simply refers to a partition $\{C_i\}_{i=1}^r$ such that $\bigcup^{r}_{i=1}C_i=[N]$, i.e. a partition by $r$ distinct classes.

Consider the following two formulations of van der Waerden, both slightly different than the originally stated formulations of van der Waerden:

The following proof is adapted from Theorem 2.4 in Landman & Robertson's Ramsey Theory on the Integers: