Some new symmetric structures in Ramsey theory

In this article, we will investigate several new configurations in Ramsey Theory, using the /varoastl,k-operation on the set of integers, recently introduced in earlier work. This operation is useful to study symmetric structures in the set of integers, such as monochromatic configurations of the form {x, y, x + y + xy} as one of its simplest cases. Previous studies have shown that Hindman's Theorem, van der Waerden's Theorem, and 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 obtain several new symmetric polynomial configurations as well as new linear symmetric patterns. Furthermore, we introduce two new operations on the set of non-negative integers N to derive additional new configurations. This work contributes to the ongoing development of symmetric structures within Ramsey Theory.