2 results
MAD SPECTRA
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- Journal:
- The Journal of Symbolic Logic / Volume 80 / Issue 3 / September 2015
- Published online by Cambridge University Press:
- 22 July 2015, pp. 901-916
- Print publication:
- September 2015
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The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on ω. We treat the problem to characterize those sets
${\rm {\cal A}} $ which, in some forcing extension of the universe, can be the mad spectrum. We give a complete solution to this problem under the assumption 
$\vartheta ^{ < \vartheta } = \vartheta $, where 
$\vartheta = {\rm{min}}\left( {\rm {\cal A}} \right) $.
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 6 / 01 December 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1139-1154
- Print publication:
- 01 December 2005
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- Article
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We prove that it is relatively consistent with
$ZFC$ that in any perfect Polish space, for every nonmeager set 
$A$ there exists a nowhere dense Cantor set 
$C$ such that 
$A\,\cap \,C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.
