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

Lu Liu; Ludovic Patey

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

Logic 2026-07-20 v1 Combinatorics

Authors: Lu Liu , Ludovic Patey

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.

Cite

@article{arxiv.2607.17666,
  title  = {Hindman's theorem does not code $\emptyset^{(\omega)}$ in one application},
  author = {Lu Liu and Ludovic Patey},
  journal= {arXiv preprint arXiv:2607.17666},
  year   = {2026}
}

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19 pages