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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04502 |
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| _version_ | 1866915314263916544 |
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| author | Ghosh, Arkabrata Goswami, Sayan Patra, Sourav Kanti |
| author_facet | Ghosh, Arkabrata Goswami, Sayan Patra, Sourav Kanti |
| contents | The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\neq x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-Čech Compactifications of discrete semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04502 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The interplay between additive and symmetric large sets and their combinatorial applications Ghosh, Arkabrata Goswami, Sayan Patra, Sourav Kanti Combinatorics The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic results. In this paper Di Nasso asked if his method could be adapted to find new non-linear Diophantine equations that are partition regular [7,Final remarks (4)]. By analyzing additive, multiplicative, and symmetric large sets, we construct new partition regular equations that give a first affirmative answer to this question. A special case of our result shows that if $P$ is a polynomial with no constant term then the equation $x+P(y-x)=z+w+zw$, where $y\neq x$ is partition regular. Also we prove several new monochromatic patterns involving additive, multiplicative, and symmetric structures. Throughout our work, we use tools from the Algebra of the Stone-Čech Compactifications of discrete semigroups. |
| title | The interplay between additive and symmetric large sets and their combinatorial applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.04502 |