Finding Product and Sum Patterns in Non-commutative Settings

Abstract

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.

Keywords

Semiring / Hindman’s Conjecture / ultrafilter / 05C55 / 37B20 / 54D80

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Peking University