Further generalization of central sets theorem for partial semigroups and vip systems

TL;DR

This paper extends the Central Sets Theorem, a key result in Ramsey theory, to broader algebraic structures called partial semigroups and VIP systems, enhancing its applicability in combinatorics and dynamics.

Contribution

It provides a new generalization of the Central Sets Theorem for arbitrary adequate partial semigroups and VIP systems, broadening the scope of previous results.

Findings

Extended the theorem to partial semigroups.

Proved the theorem for VIP systems.

Unified algebraic and topological characterizations.

Abstract

The Central Sets Theorem, a fundamental result in Ramsey theory, is a joint extension of both Hindman's theorem and van der Waerden's theorem. It was originally introduced by H. Furstenberg using methods from topological dynamics. Later, using the algebraic structure of the Stone-$\v{C}$ech compactification $\beta$ S of a semigroup S, N. Hindman and V. Bergelson extended the theorem in 1990. H. Shi and H. Yang established a topological dynamical characterization of central sets in an arbitrary semigroup (S,+), and showed it to be equivalent to the usual algebraic characterization. D. De, N. Hindman, and D. Strauss later proved a stronger version of the Central Sets Theorem for semigroups in 2008. D. Phulara further genaralized the result for commutative semigroups in 2015. Recently in his work, Zhang generalized it further and proved the central sets theorem for uncountably many central sets. We extend the theorem to arbitrary adequate partial semigroups and VIP systems.