Monochromatic exponential triples: an ultrafilter proof

2022/07/24 by Mauro Di Nasso, Di Nasso, Mauro, Mariaclara Ragosta +1 · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Mathematical Dynamics and Fractals #math.CO #msc:05A17 #msc:05D10 #msc:37B20 #msc:54D80

paper · pdf · doi:10.48550/arxiv.2207.11764

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openalex publication_date 2022/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30 · arxiv created 2026/07/31 · arxiv updated 2026/08/03

Abstract

We present a short ultrafilter proof of the existence of monochromatic exponential triples \a, b, b^a\ in any finite coloring of the natural numbers. The proof is given from scratch and uses only Ramsey's theorem, the notion of asymptotic density and the definition of ultrafilter as prerequisites. We then generalize the construction using a special ultrafilter whose existence is well known in the algebra of ultrafilters, and prove a new result on the existence of infinite monochromatic exponential patterns.

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