A 2-coloring of can have monochromatic Schur triples, but not less.
A Banach space dichotomy theorem for quotients of subspaces
A Banach space X with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if X/Y is hereditarily indecomposable for any infinite-codimensional subspace Y with a successive finite-dimensional decomposition on the basis of X. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.
A big symmetric planar set with small category projections
We show that under appropriate set-theoretic assumptions (which follow from Martin's axiom and the continuum hypothesis) there exists a nowhere meager set A ⊂ ℝ such that (i) the set {c ∈ ℝ: π[(f+c) ∩ (A×A)] is not meager} is meager for each continuous nowhere constant function f: ℝ → ℝ, (ii) the set {c ∈ ℝ: (f+c) ∩ (A×A) = ∅} is nowhere meager for each continuous function f: ℝ → ℝ. The existence of such a set also follows from the principle CPA, which...
A binary intuitionistic fuzzy relation: some new results, a general factorization, and two properties of strict components.
A Cantor-Bendixson theorem for the space
A cardinal preserving immune partition of the ordinals
A categorical characterization of sets among classes
A Čech function in ZFC
A nontrivial surjective Čech closure function is constructed in ZFC.
A characterization of determinacy for Turing degree games
A characterization of Ext(G,ℤ) assuming (V = L)
We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals satisfying (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that equals the p-rank of Ext(G,ℤ) for every...
A characterization of lifting generics for Sacks-like forcings
A characterization of polarities whose lattice of polars is Boolean
A characterization of the meager ideal
We give a classical proof of the theorem stating that the -ideal of meager sets is the unique -ideal on a Polish group, generated by closed sets which is invariant under translations and ergodic.
A characterization of tribes with respect to the Łukasiewicz -norm
We give a complete characterization of tribes with respect to the Łukasiewicz -norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the Łukasiewicz -norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental -norms, e. g., for the product -norm.
A characterization of uninorms on bounded lattices via closure and interior operators
Uninorms on bounded lattices have been recently a remarkable field of inquiry. In the present study, we introduce two novel construction approaches for uninorms on bounded lattices with a neutral element, where some necessary and sufficient conditions are required. These constructions exploit a t-norm and a closure operator, or a t-conorm and an interior operator on a bounded lattice. Some illustrative examples are also included to help comprehend the newly added classes of uninorms.
A characterization of well-orders
A class of -preduals which are isomorphic to quotients of
For every countable ordinal α, we construct an -predual which is isometric to a subspace of and isomorphic to a quotient of . However, is not isomorphic to a subspace of .
A class of sets whose distance set fills an interval
A classification of definable forcings on ω1
Under the assumption of the existence of sharps for reals all simply definable posets on are classified up to forcing equivalence.
A classification of ordinals up to Borel isomorphism
We consider the Borel structures on ordinals generated by their order topologies and provide a complete classification of all ordinals up to Borel isomorphism in ZFC. We also consider the same classification problem in the context of AD and give a partial answer for ordinals ≤ω₂.