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]
References
- Harry, Furstenberg (July 2014). Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey. ISBN 9780691615363. OCLC 889248822.
- 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.
- 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.
- 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.
- 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. S2CID 11437903.
- Hindman, Neil; Strauss, Dona (1998). Algebra in the Stone-Čech compactification : theory and applications. New York: Walter de Gruyter. ISBN 311015420X. OCLC 39368501.
- 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 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. Anal. Math. 34 (1978), pp. 61–85
- J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 317–332