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Thread: A proof using Van der Waerden's Theorem

  1. #1
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    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.
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  2. #2
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    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 $\displaystyle B$.
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