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Discrete n-tuples in Hausdorff spaces

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Carlson T., Hindman N., Strauss D.
Fundamenta Mathematicae
|
2005
|
tom 187
|
nr 2
111-126
EN
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 which are the limit of injective sequences". The answers to the remaining instances involve the notion of n-Ramsey limit. We also show that the class of spaces satisfying these discreteness conclusions for all n includes the class of F-spaces. In particular, it includes the Stone-Čech compactification of any discrete space.
2
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Longer chains of idempotents in βG

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Hindman N., Strauss D., Zelenyuk Y.
Fundamenta Mathematicae
|
2013
|
tom 220
|
nr 3
243-261
EN
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. (It had been an open question whether there was a strictly decreasing chain $⟨q_{σ}⟩_{σ<ω+1}$ in ℕ*.) As other corollaries we show that if S is an infinite right cancellative and weakly left cancellative discrete semigroup, then βS contains a decreasing chain of idempotents of reverse order type λ for every countable ordinal λ and that if S is an infinite cancellative semigroup then the set U(S) of uniform ultrafilters contains such decreasing chains.
3
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A new and stronger central sets theorem

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De D., Hindman N., Strauss D.
Fundamenta Mathematicae
|
2008
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tom 199
|
nr 2
155-175
EN
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 of the Central Sets Theorem for arbitrary semigroups S which applies to all sequences in S at once. We show that the new version is strictly stronger than the original version applied to the semigroup (ℝ,+). And we show that the noncommutative versions are strictly increasing in strength.
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