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<p>Hindman conjectured that for any finite partition of ℕ, there exists a monochromatic set of the form {<i>x,y,x</i> + <i>y,xy</i>}. Recently, Bowen proved this conjecture for all 2-partitions. In this paper, we extend Bowen’s result to semirings (<i>S</i>, +, ·) where <i>S</i> · <i>s</i> is piecewise syndetic for every <i>s</i> ∈ <i>S</i>. To achieve this, we provide a combinatorial proof of a piecewise syndetic generalization of the Bergelson–Glasscock <b>IP</b><Stack> <sub><i>r</i></sub> <sup>*</sup> </Stack> van der Waerden’s Theorem. Furthermore, we address the non-commutative case, discussing extensions to semirings with non-commutative operations.</p>

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Finding Product and Sum Patterns in Non-commutative Settings

  • 期刊论文

  • 发表日期：

    2025年7月15日

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  • Tianyi Tao,
  • Ningyuan Yang

摘要

Hindman conjectured that for any finite partition of ℕ, there exists a monochromatic set of the form {x,y,x + y,xy}. Recently, Bowen proved this conjecture for all 2-partitions. In this paper, we extend Bowen’s result to semirings (S, +, ·) where S · s is piecewise syndetic for every s ∈ S. To achieve this, we provide a combinatorial proof of a piecewise syndetic generalization of the Bergelson–Glasscock IP r * van der Waerden’s Theorem. Furthermore, we address the non-commutative case, discussing extensions to semirings with non-commutative operations.