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1
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MAD families with strong combinatorial properties

100%
Brendle J., Piper G.
Fundamenta Mathematicae
|
2007
|
tom 193
|
nr 1
7-21
EN
In his paper in Fund. Math. 178 (2003), Miller presented two conjectures regarding MAD families. The first is that CH implies the existence of a MAD family that is also a σ-set. The second is that under CH, there is a MAD family concentrated on a countable subset. These are proved in the present paper.
2
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Forcing tightness in products of fans

100%
Brendle J., Berge T. L.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 3
211-226
EN
We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.
3
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Rothberger gaps in fragmented ideals

100%
Brendle J., Mejía D.
Fundamenta Mathematicae
|
2014
|
tom 227
|
nr 1
35-68
EN
The Rothberger number 𝔟(ℐ) of a definable ideal ℐ on ω is the least cardinal κ such that there exists a Rothberger gap of type (ω,κ) in the quotient algebra 𝓟(ω)/ℐ. We investigate 𝔟(ℐ) for a class of $F_{σ}$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is ℵ₁, while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.
4
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Coloring ordinals by reals

100%
Brendle J., Fuchino S.
Fundamenta Mathematicae
|
2007
|
tom 196
|
nr 2
151-195
EN
We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^{s}(κ)$ and $F^{s}(κ)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂)) holds in a generic extension by countable support side-by-side product of Sacks or Prikry-Silver forcing (Corollary 4.8). We also show that the latter result is optimal (Theorem 5.2). Relations between these principles and their influence on the values of the variations $𝔟^{↑}$, $𝔟^{h}$, 𝔟*, 𝔡𝔬 of the bounding number 𝔟 are studied. One of the consequences of HP(κ) besides $C^{s}(κ)$ is that there is no projective well-ordering of length κ on any subset of $^{ω}ω$. We construct a model in which there is no projective well-ordering of length ω₂ on any subset of $^{ω}ω$ (𝔡𝔬 = ℵ₁ in our terminology) while 𝔟* = ℵ₂ (Theorem 6.4).
5
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Strolling through paradise

35%
Brendle J.
Fundamenta Mathematicae
|
1995
|
tom 148
|
nr 1
1-25
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