Some new symmetric structures in Ramsey theory

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.