IP set

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In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.

The finite sums shut up 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 gay number of IP sets. They require that FS(D) equal A instead of just being a subset.

The name IP-set was coined by Furstenberg and Weiss to abbreviate "Infinite-dimensional Parallelepiped".

Contents 1 Hindman's Theorem 2 Semigroups 3 See also 4 References

[edit] 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.

Since the set of natural numbers itself is an IP-set and partitions can also be seen as colorings, we 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.

[edit] 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.

[edit] See also

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

[edit] References

• V. Bergelson, I. J. H. Knutson, R. McCutcheon "Simultaneous diophantine approximation and VIP Systems" Acta Arith. 116, Academia Scientiarum Polona, (2005), 13-23
• V. 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)
• V. Bergelson, N. Hindman, "Partition regular structures contained in large sets are abundant" J. Comb. Theory (Series A) 93 (2001), pp. 18-36
• 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