IP set

In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.

The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D. The set of all finite sums over D is often denoted as FS(D).

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.

Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset.

Sources disagree on the origin of the name IP set. Some claim it was coined by Furstenberg and Weiss to abbreviate "infinite-dimensional parallelepiped", while others claim that it abbreviates "idempotent" (since a set is IP if and only if it is a member of an idempotent ultrafilter).

If ${\displaystyle S\,}$ is an IP set and ${\displaystyle S=C_{1}\cup C_{2}\cup ...\cup C_{n}}$, then at least one ${\displaystyle C_{i}\,}$ contains an IP set. This is known as Hindman's theorem or the finite sums theorem.

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