Abstract:We show that for any finite partition of $\mathbb{N}$ there is an infinite sequence whose finite sums are monochromatic and such that infinitely many of the products with a fixed number of factors are monochromatic -- though not necessarily belonging to the same color class as the finite sums. We are able to build these infinite configurations in parallel by refining arbitrary partitions of $\mathbb{N}$. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of $\mathbb{N}$.
| Comments: | 21 pages |
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.24751 [math.CO] |
| (or arXiv:2605.24751v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24751 arXiv-issued DOI via DataCite (pending registration) |
From: Conner Griffin [view email]
[v1]
Sat, 23 May 2026 22:03:33 UTC (18 KB)
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