Beyond the Galvin--Hindman--Furstenberg Theorem:\\ Partition regularity of IP sets in arbitrary semigroups

Abstract: According to Furstenberg, an \emph{infinite parallelepiped (IP)} set in a semigroup is a set that contains a sequence $a_1,a_2,\dotsc$, together with all finite sums $a_{i_1}+ \dotsb + a_{i_m}$, for natural numbers $m$ and $i_1< \dotsb <i_m$. Using a method of Galvin, Hindman proved that IP sets of natural numbers are \emph{partition regular}: For each finite partition of an IP set, some part must be an IP set, too. Furstenberg noted that the same proof applies to arbitrary semigroups. However, in a semigroup with idempotents, an IP set may be a singleton. A \emph{proper IP set} is one where the witnessing sequence $a_1,a_2,\dotsc$ may be chosen to be \emph{bijective} (in the naturals, every IP set is proper). We provide a complete characterization of semigroups where the proper IP sets are partition regular, and show that this property is equivalent to several other fundamental notions of additive Ramsey theory. Using our results, we prove that if there is a monochromatic proper IP set for each finite coloring of a semigroup, then for each finite coloring of the semigroup there are infinitely many, pairwise disjoint, proper IP sets sharing the same color.