Rigid Boolean Algebras
Remarks on cellularity in products
Rigid Aronszajn Trees
Very Strongly Rigid Boolean Algebras
The functor σ²X
We disprove the existence of a universal object in several classes of spaces including the class of weakly Lindelöf Banach spaces.
Remarks on chain conditions in products
Gaps in analytic quotients
We prove that the quotient algebra P(ℕ)/I over any analytic ideal I on ℕ contains a Hausdorff gap.
Analytic gaps
We investigate when two orthogonal families of sets of integers can be separated if one of them is analytic.
A dichotomy for P-ideals of countable sets
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 / but also on some higher order statements like for example the existence of Jensen square sequences.
A problem of von Neumann and Maharam about algebras supporting continuous submeasures
We show that a σ-algebra 𝔹 carries a strictly positive continuous submeasure if and only if 𝔹 is weakly distributive and it satisfies the σ-finite chain condition of Horn and Tarski.
The isomorphism relation between tree-automatic Structures
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...
Partition properties of ω1 compatible with CH
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.
Chains and antichains in Boolean algebras
We give an affirmative answer to problem DJ from Fremlin’s list [8] which asks whether implies that every uncountable Boolean algebra has an uncountable set of pairwise incomparable elements.
Cofinal types of topological directed orders
We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.
Chain conditions in maximal models
We present two 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 ₃ fails. Of particular interest is the still open question...
Multiple gaps
We study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.
Rectangular square-bracket operation for successor of regular cardinals
We give a uniform proof that holds for every regular cardinal λ.
A universal sequence of continuous functions
Bounded sets in topological vector spaces.
Partial unconditionality of weakly null sequences.
We survey a combinatorial framework for studying subsequences of a given sequence in a Banach space, with particular emphasis on weakly-null sequences. We base our presentation on the crucial notion of introduced long time ago by Nash-Williams. In fact, one of the purposes of this survey is to isolate the importance of studying mappings defined on barriers as a crucial step towards solving a given problem that involves sequences in Banach spaces. We focus our study on various forms of ?partial...
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