Partition regularity of infinite parallelepiped sets
Combinatorics
2026-01-13 v2
Abstract
A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence 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.
Keywords
Cite
@article{arxiv.2212.06887,
title = {Partition regularity of infinite parallelepiped sets},
author = {Yonatan Gadot and Boaz Tsaban},
journal= {arXiv preprint arXiv:2212.06887},
year = {2026}
}