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A new and stronger central sets theorem

Dibyendu DeNeil HindmanDona Strauss — 2008

Fundamenta Mathematicae

Furstenberg's original Central Sets Theorem applied to central subsets of ℕ and finitely many specified sequences in ℤ. In this form it was already strong enough to derive some very strong combinatorial consequences, such as the fact that a central subset of ℕ contains solutions to all partition regular systems of homogeneous equations. Subsequently the Central Sets Theorem was extended to apply to arbitrary semigroups and countably many specified sequences. In this paper we derive a new version...

Longer chains of idempotents in βG

Neil HindmanDona StraussYevhen Zelenyuk — 2013

Fundamenta Mathematicae

Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef. We show that if G is a countable discrete group, p is a right cancelable element of G* = βG∖G, and λ is a countable ordinal, then there is a strictly decreasing chain q σ σ < λ of idempotents in C p , the smallest compact subsemigroup of G* with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S* is the largest element of such a strictly decreasing chain of idempotents....

Discrete n-tuples in Hausdorff spaces

Timothy J. CarlsonNeil HindmanDona Strauss — 2005

Fundamenta Mathematicae

We investigate the following three questions: Let n ∈ ℕ. For which Hausdorff spaces X is it true that whenever Γ is an arbitrary (respectively finite-to-one, respectively injective) function from ℕⁿ to X, there must exist an infinite subset M of ℕ such that Γ[Mⁿ] is discrete? Of course, if n = 1 the answer to all three questions is "all of them". For n ≥ 2 the answers to the second and third questions are the same; in the case n = 2 that answer is "those for which there are only finitely many points...

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