We study the existence of Borel sets B ⊆ ω2 admitting a sequence 〈ηα : αλ〉 of distinct elements of ω2 such that |(ηα +B)∩(ηβ +B)| ≥ 6 for all α,β λ but with no perfect set of such η’s. Our result implies that under the Martin Axiom, if ℵα  c, α ω1 and 3 ≤ ι ω, then there exists a Σ0 2 set B ⊆ ω2 which has ℵα many pairwise 2ι–nondisjoint translations but not a perfect set of such translations. Our arguments closely follow Shelah [7, Section 1]

It is proved that the Tang-Pequignot reducibility (or reducibility by relatively continuous relations) on a  second countable, T₀ space X either coincides with the Wadge reducibility for the given topology, or there is no topology on X that can turn it into Wadge reducibility.

The paper first covers several properties of the extension of the divisibility relation to a set *N of nonstandard integers, including an analogue of the basic theorem of arithmetic. After that, a connection is established with the divisibility in the Stone–Čech compactification βN, proving that the divisibility of ultrafilters introduced by the author is equivalent to divisibility of some elements belonging to their respective monads in an enlargement. Some earlier results on ultrafilters on lower levels on the divisibility hierarchy are illuminated by nonstandard methods. Using limits by ultrafilters we obtain results on ultrafilters above these finite levels, showing that for them a distribution by levels is not possible.

Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we  look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.

We show that there are 0-definably complete ordered fields which are not real closed. Therefore, the theory of definably with parameters complete ordered fields does not follow from the theory of 0-definably complete ordered fields. The mentioned completeness notions for ordered fields are the definable versions of completeness in the sense of Dedekind cuts. In earlier joint work, we had shown that it would become successively weakened if we just required nonexistence of definable regular gaps and then disallowing parameters. The result in this note shows reducing in the opposite order, at least one side is sharp.

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. A point-line geometry is a non-empty set of elements called points, together with a collection of subsets, called lines, in a way that every line contains at least two points and any pair of points is contained in at most one line. A line which contains more than two points is called a regular line. Point-line geometries can alternatively be formalised as relational structures. We establish a correspondence between the point-line geometries investigated in this paper and the firstorder structures with a single ternary relation L satisfying certain axioms (i.e. that the class of point-line geometries corresponds to a subclass of 3-uniform hypergraphs). We characterise the homomorphism-homogeneous point-line geometries with two regular non-intersecting lines. Homomorphism-homogeneous pointline geometries containing two regular intersecting lines have already been classified by Mašulović.

In this work, we consider two families of incidence problems, C₁ and C₂,, which are related to real numbers and countable subsets of the real line. Instances of problems of C₁ are as follows: given a real number x, pick randomly a countable set of reals A hoping that x ∈ A, whereas instances of problems of C₂ are as follows: given a countable set of reals A, pick randomly a real number x hoping that x ∉ A. One could arguably defend that, at least intuitively, problems of C₂ are easier to solve than problems of C₁. After some suitable formalization, we prove (within ZFC) that, on one hand, problems of C₂ are, indeed, at least as easy to solve as problems of C₁. On the other hand, the statement “Problems of C₁ have the exact same complexity of problems of C₂” is shown to be an equivalent of the Continuum Hypothesis.