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).

Hindman's theorem

If $S\,$ 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.

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.

Semigroups

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.

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Syndetic set
• Thick set

References

• Vitaly Bergelson, I. J. H. Knutson, R. McCutcheon "Simultaneous diophantine approximation and VIP Systems" Acta Arith. 116, Academia Scientiarum Polona, (2005), 13-23
• Vitaly Bergelson, "Minimal Idempotents and Ergodic Ramsey Theory" Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
• Bergelson, V.; Hindman, N. (2001). "Partition regular structures contained in large sets are abundant". J. Comb. Theory (Series A). 93: 18–36. doi:10.1006/jcta.2000.3061. A publicly available copy is hosted by one of the authors.
• H. Furstenberg, B. Weiss, "Topological Dynamics and Combinatorial Number Theory", J. d'Analyse Math. 34 (1978), pp. 61-85
• J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 317-332