Partition regularity of Pythagorean pairs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916619505106944 |
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| author | Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel |
| author_facet | Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel |
| contents | We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10636 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Partition regularity of Pythagorean pairs Frantzikinakis, Nikos Klurman, Oleksiy Moreira, Joel Combinatorics Number Theory Primary:05D10, Secondary:11N37, 11B30, 37A44 We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions. |
| title | Partition regularity of Pythagorean pairs |
| topic | Combinatorics Number Theory Primary:05D10, Secondary:11N37, 11B30, 37A44 |
| url | https://arxiv.org/abs/2309.10636 |