Partition regularity of Pythagorean pairs
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 partitio...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508624000271/type/journal_article |
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| author | Nikos Frantzikinakis Oleksiy Klurman Joel Moreira |
| author_facet | Nikos Frantzikinakis Oleksiy Klurman Joel Moreira |
| author_sort | Nikos Frantzikinakis |
| collection | DOAJ |
| description | 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 | Article |
| id | doaj-art-b11e603cf9bb42ed879aa02b4caacd45 |
| institution | Kabale University |
| issn | 2050-5086 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-b11e603cf9bb42ed879aa02b4caacd452025-02-12T03:38:02ZengCambridge University PressForum of Mathematics, Pi2050-50862025-01-011310.1017/fmp.2024.27Partition regularity of Pythagorean pairsNikos Frantzikinakis0Oleksiy Klurman1Joel Moreira2University of Crete, Department of Mathematics and Applied Mathematics, Heraklion Greece; E-mail:School of Mathematics, University of Bristol, Bristol, UK; E-mail:Warwick Mathematics Institute, University of Warwick, Coventry, UK;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.https://www.cambridge.org/core/product/identifier/S2050508624000271/type/journal_article05D1011N3711B3037A44 |
| spellingShingle | Nikos Frantzikinakis Oleksiy Klurman Joel Moreira Partition regularity of Pythagorean pairs Forum of Mathematics, Pi 05D10 11N37 11B30 37A44 |
| title | Partition regularity of Pythagorean pairs |
| title_full | Partition regularity of Pythagorean pairs |
| title_fullStr | Partition regularity of Pythagorean pairs |
| title_full_unstemmed | Partition regularity of Pythagorean pairs |
| title_short | Partition regularity of Pythagorean pairs |
| title_sort | partition regularity of pythagorean pairs |
| topic | 05D10 11N37 11B30 37A44 |
| url | https://www.cambridge.org/core/product/identifier/S2050508624000271/type/journal_article |
| work_keys_str_mv | AT nikosfrantzikinakis partitionregularityofpythagoreanpairs AT oleksiyklurman partitionregularityofpythagoreanpairs AT joelmoreira partitionregularityofpythagoreanpairs |