IP crowd

In mathematics , the term IP set denotes a set of natural numbers that contains all finite sums of an infinite set .

The finite sums of a set of natural numbers are the numbers that can be represented as the sum of the elements of a non-empty finite subset of . The set of all finite sums of is also denoted as; FS stands for Finite Sums . ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle FS(D)}$

A lot of natural numbers is an IP set if an infinite set exists, so that in is included. ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle FS(D)}$ ${\displaystyle A}$

Sometimes a slightly different definition is used: one then demands that is even for a suitable one . ${\displaystyle A=FS(D)}$ ${\displaystyle D}$

The designation IP set (IP set) goes back to Hillel Fürstenberg and Barak Weiss ; IP stands for "Infinite-dimensional Parallelepiped ".

Hindman's theorem

Hindman's theorem, or the Finite Sums Theorem , is as follows:

If an IP set is and , then at least one of the sets is an IP set. ${\displaystyle S\,}$ ${\displaystyle S=C_{1}\cup C_{2}\cup ...\cup C_{n}}$ ${\displaystyle C_{i}\,}$

Since the set of natural numbers is itself also an IP set and partitions can also be understood as colorations, the following special case of Hindman's theorem can be formulated:

If the natural numbers are colored, then there is an infinite quantity for at least one color , so that all elements of and even all finite sums of have the color . ${\displaystyle n}$ ${\displaystyle c}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle c}$

Semigroups

The IP property can be defined not only for the natural numbers that form a semigroup when added, but also in general for semigroups and partial semigroups.

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• V. Bergelson, IJH Knutson, R. McCutcheon: Simultaneous diophantine approximation and VIP Systems (PDF; 127 kB) Acta Arith. 116 , Academia Scientiarum Polona, ​​(2005), 13-23
• V. Bergelson: Minimal Idempotents and Ergodic Ramsey Theory (PDF; 349 kB) Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310 , Cambridge Univ. Press, Cambridge, (2003)
• H. Fürstenberg , B. Weiss: Topological Dynamics and Combinatorial Number Theory, J. d'Analyse Math. 34 (1978), 61-85
• J. McLeod: Some Notions of Size in Partial Semigroups (PDF; 164 kB) Topology Proceedings, Vol. 25 (2000), 317-332