Some new symmetric structures in Ramsey theory

TL;DR

This paper explores new symmetric configurations in Ramsey theory using a novel operation on integers, including polynomial and linear patterns, and introduces two new operations to generate further configurations.

Contribution

It introduces new symmetric structures in Ramsey theory, including polynomial and linear configurations, and proposes two new operations on integers for generating additional patterns.

Findings

Discovered new symmetric polynomial configurations.

Identified new linear symmetric patterns.

Introduced two novel operations on non-negative integers.

Abstract

In this article, we will investigate several new configurations in Ramsey Theory, using the $\ostar_{l,k}$-operation on the set of integers, recently introduced in \cite{key-4}. This operation is useful to study symmetric structures in the set of integers, such as monochromatic configurations of the form $\left\{ x,y,x+y+xy\right\} $ as one of its simplest case. In \cite{key-4}, the author has studied more general symmetric structures. It has been shown that the Hindman's Theorem, van der Waerden's Theorem, Deuber's Theorem have their own symmetric versions. In this article we will explore several new structures, including polynomial versions of these symmetric structures and some of its variants. As a result, we get several new symmetric polynomial configurations as well as new linear symmetric patterns. In the final section, we will also introduce two new operations on the set of non-negative integers $\mathbb{N}$, to obtain further new configurations.