Non commutative and polynomial Multidimensional Stronger Central Sets Theorem

Abstract

Hindman's theorem and van der Waerden's theorem are
two classical Ramsey theoretic results, the first one deals with infinite
configurations and the second one deals with finite configurations.
The Central Sets Theorem due to Furstenberg is a strong simultaneous
extension of both theorems, which also applies to general commutative
semigroups. Beiglboeck provided a common extension of the Central
Sets Theorem and Milliken-Taylor Theorem in commutative case. Furstenberg's
original Central Sets Theorem was proved in \cite{key-2}
for finitely many sequences in $\left(\mathbb{N},+\right)$ at a time. Bergelson and Hindman provided
a non commutative version of this Theorem \cite{key-3}. Hindman,
Maleki and Strauss extended the Central Sets Theorem for commutative
semigroup using countable many sequences at a time \cite{key-8-1}.
De, Hindman and Strauss provided a non-commutative version of Central Sets Theorem using arbitrary
many sequences at a time \cite{key-5}. In this work we will provide
a non-commutative extension of Beiglboeck's Theorem. We also provide
polynomial generalization of Beiglboeck's theorem.