Partition regularity of infinite parallelepiped sets
Yonatan Gadot, Boaz Tsaban
TL;DR
The paper provides a complete classification of semigroups in which proper IP sets are partition regular, showing this property holds precisely when no subsemigroup falls into three explicit obstruction types. It introduces and analyzes proper sumsequences to characterize when injective sequences yield proper IP sets, linking these conditions to core additive Ramsey principles. The authors also show a negative answer to a question by Andrews and Goldbring by constructing a non-moving semigroup with partition-regular proper IP sets, and they prove a stronger Hindman-type result: every finite coloring yields infinitely many pairwise disjoint monochromatic proper IP sets. Together, these results unify several Ramsey-theoretic notions in semigroups under a sharp obstruction-based criterion.
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.