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). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni).

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 Furstenberg and Weiss[1] to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent"[2] (a set is IP if and only if it is a member of an idempotent ultrafilter).

Hindman's theorem

If is an IP set and , then at least one is an IP set. This is known as Hindman's theorem or the finite sums theorem.[3][4] In different terms, Hindman's theorem states that the class of IP sets is partition regular.

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.

The Milliken–Taylor theorem is a common generalisation of Hindman's theorem and Ramsey's theorem.

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. A variant of Hindman's theorem is true for arbitrary semigroups.[5][6]

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gollark: That sounds very cool if quite possibly impractical.
gollark: There aren't that many alternatives.
gollark: Personally, my suggested climate-change-handling policies:- massively scale up nuclear fission power, it's just great in most ways- invest in better rail infrastructure - maglevs are extremely cool™ and fast™ and could maybe partly replace planes?- electric cars could be rented from a local "pool" for intra-city transport, which would save a lot of cost on batteries- increase grid interconnectivity so renewables might be less spotty- impose taxes on particularly badly polluting things- do research into geoengineering things which can keep the temperature from going up as much- increase standards for reparability; we lose so many resources to randomly throwing stuff away because they're designed with planned obsolecence- a very specific thing related to that bit above there - PoE/other low-voltage power grids in homes, since centralizing all the AC→DC conversion circuitry could improve efficiency, lower costs of end-user devices, and make LED lightbulbs less likely to fail (currently some of them include dirt-cheap PSUs which have all *kinds* of problems)

See also

References
  1. Harry, Furstenberg. Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey. ISBN 9780691615363. OCLC 889248822.
  2. Bergelson, V.; Leibman, A. (2016). "Sets of large values of correlation functions for polynomial cubic configurations". Ergodic Theory and Dynamical Systems. 38 (2): 499–522. doi:10.1017/etds.2016.49. ISSN 0143-3857.
  3. Hindman, Neil (1974). "Finite sums from sequences within cells of a partition of N". Journal of Combinatorial Theory, Series A. 17 (1): 1–11. doi:10.1016/0097-3165(74)90023-5. hdl:10338.dmlcz/127803.
  4. Baumgartner, James E (1974). "A short proof of Hindman's theorem". Journal of Combinatorial Theory, Series A. 17 (3): 384–386. doi:10.1016/0097-3165(74)90103-4.
  5. Golan, Gili; Tsaban, Boaz (2013). "Hindmanʼs coloring theorem in arbitrary semigroups". Journal of Algebra. 395: 111–120. arXiv:1303.3600. doi:10.1016/j.jalgebra.2013.08.007.
  6. Hindman, Neil; Strauss, Dona (1998). Algebra in the Stone-Čech compactification : theory and applications. New York: Walter de Gruyter. ISBN 311015420X. OCLC 39368501.
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