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Infinite Sum-Product Configurations in Parallel
Abstract
We show that for any finite partition of 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 ℕ. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of ℕ.
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BibTeX
@misc{griffin2026infinite,
title = {{Infinite Sum-Product Configurations in Parallel}},
author = {Conner Griffin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.24751},
eprint = {2605.24751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.24751},
}
RIS
TY - UNPB
TI - Infinite Sum-Product Configurations in Parallel
AU - Conner Griffin
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2605.24751
UR - https://arxiv.org/abs/2605.24751
AB - We show that for any finite partition of 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 ℕ. We apply these techniques to prove that many complex infinite sum-product configurations are guaranteed to be monochromatic for arbitrary finite colorings of ℕ.
ER -