Hindman's theorem does not code $\emptyset^{(\omega)}$ in one application

Lu Liu; Ludovic Patey

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arXiv · 2607.17666

Hindman's theorem does not code $\emptyset^{(\omega)}$ in one application

Lu Liu·Ludovic Patey

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Abstract

We prove that for every non-arithmetic set~$C$ and every arithmetic finite coloring of~$\mathbb{N}$, there is an infinite set $H \subseteq \mathbb{N}$ whose non-empty finite sums of distinct elements is monochromatic, and $C$ is not $H$-computable. We also study restrictions of Hindman's theorem to simple colorings.

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Lu Liu, Ludovic Patey. 2026-07-20. Hindman's theorem does not code $\emptyset^{(\omega)}$ in one application. https://arxiv.org/abs/2607.17666

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