4,177 research outputs found
Maximal almost disjoint families, determinacy, and forcing
We study the notion of -MAD families where is a
Borel ideal on . We show that if is an arbitrary
ideal, or is any finite or countably iterated Fubini product of
ideals, then there are no analytic infinite -MAD
families, and assuming Projective Determinacy there are no infinite projective
-MAD families; and under the full Axiom of Determinacy +
there are no infinite -mad families.
These results apply in particular when is the ideal of finite sets
, which corresponds to the classical notion of MAD families. The
proofs combine ideas from invariant descriptive set theory and forcing.Comment: 40 page
The Ramsey property implies no mad families
We show that if all collections of infinite subsets of have the Ramsey
property, then there are no infinite maximal almost disjoint (mad) families.
This solves a long-standing problem going back to Mathias \cite{mathias}. The
proof exploits an idea which has its natural roots in ergodic theory,
topological dynamics, and invariant descriptive set theory: We use that a
certain function associated to a purported mad family is invariant under the
equivalence relation , and thus is constant on a "large" set. Furthermore
we announce a number of additional results about mad families relative to more
complicated Borel ideals.Comment: 10 pages; fixed a mistake in Theorem 4.
Projective maximal families of orthogonal measures with large continuum
We study maximal orthogonal families of Borel probability measures on
(abbreviated m.o. families) and show that there are generic
extensions of the constructible universe in which each of the following
holds:
(1) There is a -definable well order of the reals, there is a
-definable m.o. family, there are no -definable
m.o. families and (in fact any reasonable
value of will do).
(2) There is a -definable well order of the reals, there is a
-definable m.o. family, there are no -definable
m.o. families, and .Comment: 12 page
and mad families
We answer in the affirmative the following question of J\"org Brendle: If
there is a mad family, is there then a mad family?Comment: This will appear in the Journal of Symbolic Logic (March 2014
Definable maximal discrete sets in forcing extensions
Let be a binary relation, and recall that a set
is -discrete if no two elements of are related by .
We show that in the Sacks and Miller forcing extensions of there is a
maximal -discrete set. We use this to answer in the
negative the main question posed in [5] by showing that in the Sacks and Miller
extensions there is a maximal orthogonal family ("mof") of Borel
probability measures on Cantor space. A similar result is also obtained for
mad families. By contrast, we show that if there is a Mathias real
over then there are no mofs.Comment: 16 page
Definable maximal cofinitary groups of intermediate size
Using almost disjoint coding, we show that for each
consistently ,
where is witnessed by a maximal cofinitary
group.Comment: 22 page
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