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Selctive ultrafilters and homogeneity. (English) Zbl 0649.03036
This paper consists, roughly speaking, of three parts: in the first part the author refines older results on the existence of homogeneous sets for partitions of \([\omega]^{\omega}\). An interesting byproduct of these investigations is the result that: if every ccc forcing adds either a Cohen or a random real, then there are no P-points. This sheds some light on the question of Laver and Prikry whether the if-clause is consistent with ZFC.
The second part investigates what happens after Lévy-collapsing a Mahlo cardinal to \(\omega_ 1\). In that model non-isomorphic selective ultrafilters are radically different: they are mutually generic on \([\omega]^{\omega}\) over the class of sets hereditarily definable over the ground model and \({\mathbb{R}}.\)
In the third part, the author shows that under CH there is a group of autohomeomorphisms of \(\beta\omega\setminus \omega\) that distinguishes non-isomorphic ultrafilters. It is a result of W. Rudin [Duke Math. J. 23, 409-419 (1956; Zbl 0073.396)] that under CH any two P-points can be mapped onto each other by an autohomeomorphism of \(\beta\omega\setminus \omega\).
Reviewer: K.P.Hart

MSC:
03E05 Other combinatorial set theory
03E55 Large cardinals
03E35 Consistency and independence results
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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