A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A. Equivalently, one may require that A contains all finite sums FS((ni)) of a sequence (ni).
Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset.
The term IP set was coined by Hillel Furstenberg and Benjamin Weiss[1][2] to abbreviate "infinite-dimensional parallelepiped".
[4][5] In different terms, Hindman's theorem states that the class of IP sets is partition regular.
A variant of Hindman's theorem is true for arbitrary semigroups.