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 (n_i), one can consider the set of finite sums FS((n_i)), consisting of the sums of all finite length subsequences of (n_i).

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((n_i)) of a sequence (n_i).

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 Hillel Furstenberg and Benjamin Weiss{{cite journal |last1=Furstenberg |first1=H. |author-link=Hillel Furstenberg |last2=Weiss |first2=B. |author-link2=Benjamin Weiss |date=December 1978 |title=Topological Dynamics and Combinatorial Number Theory |journal=Journal d'Analyse Mathématique |volume=34 |pages=61–85 |doi=10.1007/BF02790008 |issn=0021-7670 |eissn=1565-8538 |doi-access=free}}{{Cite book |last=Furstenburg |first=Harry |title=Recurrence in ergodic theory and combinatorial number theory |publisher=Princeton University Press |year=1981 |isbn=9781400855162 |location=Princeton, New Jersey |oclc=889248822}} to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent"{{Cite journal |last1=Bergelson |first1=V. |last2=Leibman |first2=A. |date=2016 |title=Sets of large values of correlation functions for polynomial cubic configurations |journal=Ergodic Theory and Dynamical Systems |publisher=Cambridge University Press |publication-date=April 2018 |volume=38 |issue=2 |pages=499–522 |doi=10.1017/etds.2016.49 |issn=0143-3857 |eissn=1469-4417 |s2cid=31083478}} (a set is an IP if and only if it is a member of an idempotent ultrafilter).

Hindman's theorem

If $S$ is an IP set and $S = C_1 \cup C_2 \cup \cdots \cup C_n$ , then at least one $C_i$ is an IP set.

This is known as Hindman's theorem or the finite sums theorem.{{Cite journal |last=Hindman |first=Neil |author-link=Neil Hindman |date=July 1974 |title=Finite sums from sequences within cells of a partition of N |journal=Journal of Combinatorial Theory |series=Series A |volume=17 |issue=1 |pages=1–11 |doi=10.1016/0097-3165(74)90023-5 |issn=0097-3165 |eissn=1096-0899 |doi-access=free |hdl-access=free |hdl=10338.dmlcz/127803}}{{Cite journal |last=Baumgartner |first=James E. |author-link=James Earl Baumgartner |date=November 1974 |title=A short proof of Hindman's theorem |journal=Journal of Combinatorial Theory |series=Series A |volume=17 |issue=3 |pages=384–386 |doi=10.1016/0097-3165(74)90103-4 |issn=0097-3165 |eissn=1096-0899 |doi-access=free}} 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 color. 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 is named for mathematician Neil Hindman, who proved it in 1974.

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.{{Cite journal |last1=Golan |first1=Gili |last2=Tsaban |first2=Boaz |author-link2=Boaz Tsaban |date=1 December 2013 |title=Hindmanʼs coloring theorem in arbitrary semigroups |journal=Journal of Algebra |publisher=Academic Press |volume=395 |pages=111–120 |arxiv=1303.3600 |doi=10.1016/j.jalgebra.2013.08.007 |issn=0021-8693 |eissn=1090-266X |s2cid=11437903 |doi-access=free}}{{Cite book |last1=Hindman |first1=Neil |author-link1=Neil Hindman |title=Algebra in the Stone-Čech Compactification: Theory and Applications |last2=Strauss |first2=Dona |author2-link=Dona Strauss |publisher=Walter de Gruyter |year=1998 |isbn=311015420X |location=New York |doi=10.1515/9783110809220 |oclc=39368501}}

IP set

Hindman's theorem

Semigroups

See also

References

Further reading