A proof using Van der Waerden's Theorem

**grad444** · September 7th 2009, 11:54 AM

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

**Taluivren** · September 8th 2009, 11:47 AM

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

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