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[who?] claim that it abbreviates "idempotent" (since a set is IP if and only if it is a member of an idempotent ultrafilter).
If is an IP set and , then at least one contains an IP set. This is known as Hindman's theorem or the finite sums theorem.
Since the set of natural numbers itself is an IP set and partitions can also be seen as colorings, one can reformulate a special case of Hindman's theorem in more familiar terms: Suppose the natural numbers are "colored" with n different colors; each natural number gets one and only one of the n colors. Then there exists a color c and an infinite set D of natural numbers, all colored with c, such that every finite sum over D also has color c.
Hindman's theorem states that the class of IP sets is partition regular.
The definition of being IP has been extended from subsets of the special semigroup of natural numbers with addition to subsets of semigroups and partial semigroups in general.
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