The interplay between additive and symmetric large sets and their combinatorial applications

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...

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Bibliographic Details
Published in:arXiv.org 2024-04
Main Authors: Ghosh, Arkabrata, Goswami, Sayan, Patra, Sourav Kanti
Format: Article
Language:English
Subjects:
Combinatorial analysis
Diophantine equation
Mathematical analysis
Online Access:Get full text
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Summary: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.
ISSN:2331-8422