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1
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Ramseyan ultrafilters

100%
Halbeisen L.
Fundamenta Mathematicae
|
2001
|
tom 169
|
nr 3
233-248
EN
We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on the set of partition-filters and consider the dual form of some cardinal characteristics of the continuum.
2
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On the complexity of Hamel bases of infinite-dimensional Banach spaces

100%
Halbeisen L.
Colloquium Mathematicum
|
2001
|
tom 89
|
nr 1
133-134
EN
We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.
3
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Optimal bounds for the length of rational Collatz cycles

64%
Halbeisen L., Hungerbühler N.
Acta Arithmetica
|
1996-1997
|
tom 78
|
nr 3
227-239
4
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Non-separable Banach spaces with non-meager Hamel basis

51%
Banakh T., Džamonja M., Halbeisen L.
Studia Mathematica
|
2008
|
tom 189
|
nr 1
27-34
EN
We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ 𝔠⁺; ∙ a dense non-meager Hamel basis if $|X| = κ^{ω} = 2^{κ}$ for some cardinal κ.
5
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On bases in Banach spaces

45%
Bartoszyński T., Džamonja M., Halbeisen L., Murtinová E., Plichko A.
Studia Mathematica
|
2005
|
tom 170
|
nr 2
147-171
EN
We investigate various kinds of bases in infinite-dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in $ℓ_{∞}$ as well as in separable Banach spaces.
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