2022/12/13 by Yonatan Gadot, Gadot, Yonatan, Boaz Tsaban +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2212.06887
openalex publication_date 2022/12/13 · openalex created_date 2022/12/27 · openalex updated_date 2026/07/28
A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence \myseqa 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.