# Thread: A proof using Van der Waerden's Theorem

1. ## A proof using Van der Waerden's Theorem

Suppose that A = {$\displaystyle a_1,a_2,...$} and B = {$\displaystyle b_1,b_2,...$} and that A contains arbitrarily large arithmetic progressions. Suppose there exists a K such that, for all n, $\displaystyle |b_n-a_n|\leq K$. Prove that B contains arbitrarily large arithmetic progressions.

I am having a hard time applying Van der Waerden's Theorem to the set B using only the information given. I know I am overlooking something, but I am not sure what. Thanks.

2. Let $\displaystyle m$ be a positive integer, we want to show $\displaystyle B$ contains an arithmetic progression of length $\displaystyle m$. For every $\displaystyle n=1,2,3,\ldots$ we have $\displaystyle b_n \in \{a_n-K, a_n-K+1, a_n-K+2,\ldots, a_n+K\}$ ...just think of these $\displaystyle 2K+1$ possibilities as of colours.
In $\displaystyle A$, there exists an arithmetic progression $\displaystyle C=\{c_1, c_2,\ldots, c_{W(m,2K+1)}\} \subseteq A$ of length $\displaystyle W(m,2K+1)$.
The fact that every $\displaystyle (2K+1)$-colouring of $\displaystyle \{1,2,3,\ldots, W(m,2K+1)\}$ contains a monochromatic arithmetic progression of length $\displaystyle m$ means that $\displaystyle C$ contains an arithmetic progression $\displaystyle D=\{d_1,d_2,\ldots, d_m\} \subseteq C$ such that there are $\displaystyle m$ elements of $\displaystyle B$ expressible as $\displaystyle d_1+t, d_2+t, d_3+t,\ldots, d_m+t$ for some $\displaystyle t \in \{-K,-K+1,\ldots, K\}$.
The elements $\displaystyle d_1+t, d_2+t, d_3+t,\ldots, d_m+t$ form an arithmetic progression of length $\displaystyle m$ in $\displaystyle B$.