New polynomial and multidimensional extensions of classical partition results

TL;DR

This paper extends classical partition regularity results to polynomial and multidimensional configurations in abelian groups and semigroups, using ultrafilter techniques and generalizing key theorems.

Contribution

It introduces polynomial and multidimensional extensions of Deuber's $(m,p,c)$-sets, broadening their applicability to abelian groups and semigroups.

Findings

Partition regular polynomial configurations in $\\mathbb{Z}^d$

Generalization of Deuber's results to semigroups

Polynomial version of the central sets theorem

Abstract

In the 1970s Deuber introduced the notion of $(m,p,c)$-sets in $\mathbb{N}$ and showed that these sets are partition regular and contain all linear partition regular configurations in $\mathbb{N}$. In this paper we obtain enhancements and extensions of classical results on $(m,p,c)$-sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in $\mathbb{Z}^d$. Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups. We also obtain a polynomial version of the central sets theorem of Furstenberg, extend the theory of $(m,p,c)$-systems of Deuber, Hindman and Lefmann and generalize a classical theorem of Rado regarding partition regularity of linear systems of equations over $\mathbb{N}$ to commutative semigroups.