Partition regularity of infinite parallelepiped sets

TL;DR

This paper classifies semigroups where infinite parallelepiped sets are partition regular, extending Hindman's theorem and linking it to core concepts in additive Ramsey theory.

Contribution

It provides a complete classification of semigroups with partition regular proper IP sets and establishes their equivalence to key additive Ramsey properties.

Findings

Proper IP sets are partition regular in certain semigroups.

The property is characterized by fundamental additive Ramsey notions.

The classification clarifies the scope of Hindman's theorem generalizations.

Abstract

A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence $\myseq{a}$ and its finite sums, or a superset of such a set. Hindman's theorem asserts that the proper IP sets of natural numbers are partition regular: for each finite coloring of a proper IP set of natural numbers there is a monochromatic proper IP subset. Furstenberg generalized this question to arbitrary semigroups, in which the analogous result does not hold in general. We provide a complete classification of the semigroups for which the proper IP sets are partition regular, and show that this property is equivalent to other fundamental notions of additive Ramsey theory.