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1
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A dichotomy for P-ideals of countable sets

100%
Todorčević S.
Fundamenta Mathematicae
|
2000
|
tom 166
|
nr 3
251-267
EN
A dichotomy concerning ideals of countable subsets of some set is introduced and proved compatible with the Continuum Hypothesis. The dichotomy has influence not only on the Suslin Hypothesis or the structure of Hausdorff gaps in the quotient algebra $P(\mathbb{N})$/ but also on some higher order statements like for example the existence of Jensen square sequences.
2
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Analytic gaps

100%
Todorčević S.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 1
55-66
EN
We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.
3
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The functor σ²X

100%
Todorčević S.
Studia Mathematica
|
1995
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tom 116
|
nr 1
49-57
EN
We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.
4
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Gaps in analytic quotients

100%
Todorčević S.
Fundamenta Mathematicae
|
1998
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tom 156
|
nr 1
85-97
EN
We prove that the quotient algebra P(ℕ)/I over any analytic ideal I on ℕ contains a Hausdorff gap.
5
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Chain conditions in maximal models

64%
Larson P., Todorčević S.
Fundamenta Mathematicae
|
2001
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tom 168
|
nr 1
77-104
EN
We present two $ℙ_{max}$ varations which create maximal models relative to certain counterexamples to Martin's Axiom, in hope of separating certain classical statements which fall between MA and Suslin's Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster's forcing axiom 𝒦₃ fails. Of particular interest is the still open question whether 𝒦₂ holds in this model.
6
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The isomorphism relation between tree-automatic Structures

64%
Finkel O., Todorčević S.
Open Mathematics
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2010
|
tom 8
|
nr 2
299-313
EN
An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set.
7
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Chains and antichains in Boolean algebras

64%
Losada M., Todorčević S.
Fundamenta Mathematicae
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2000
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tom 163
|
nr 1
55-76
EN
We give an affirmative answer to problem DJ from Fremlin's list [8] which asks whether $MA_{ω_1}$ implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.
8
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Partition properties of ω1 compatible with CH

64%
Abraham U., Todorčević S.
Fundamenta Mathematicae
|
1997
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tom 152
|
nr 2
165-181
EN
A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.
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