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Math Help - 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 = { 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.
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  2. #2
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    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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