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Improved Lower Bounds for Van Der Waerden Numbers


Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by \({k^{{b_0}\left( k \right)}}\) where \({b_0}\left( k \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}\). We prove a new lower bound of kb(k) with \(b\left( k \right) = {{c\log \,k} \over {\log \log \,k}}\). This is done by modifying Green’s argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.

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The author would like to thank Ben Green for many helpful comments concerning the presentation of this paper. The author also thanks Zachary Chase for his encouragement, and for checking a draft of this paper. Lastly, the author is grateful for the helpful reports of two anonymous referees.

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Correspondence to Zach Hunter.

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Hunter, Z. Improved Lower Bounds for Van Der Waerden Numbers. Combinatorica 42 (Suppl 2), 1231–1252 (2022).

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Mathematics Subject Classification (2010)

  • 05D10
  • 11B25