Related papers: Q

For any set $X$, ${\mathcal P}(X)$ denotes the collection of all subsets of $X$, ordered by inclusion. A {\it cutset} in ${\mathcal P}(X)$ is a subset of ${\mathcal P}(X)$ which meets every maximal chain of ${\mathcal P}(X)$. A cutset is…

We show that it is consistent that the continuum is as large as you wish, and for each uncountable cardinal $\kappa$ below the continuum, there are a subset $T$ of the reals and a family $A$ of countable subsets of $T$ such that (1) both…

We prove the consistency of the existence of a $Q$-set whose square is not a $\Delta$-set and that if there is a $\Delta$-set, then there exists a $\Delta$-set whose all finite powers are $\Delta$-sets.

In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is…

Given an uncountable cardinal $\kappa$, we consider the question of whether subsets of the power set of $\kappa$ that are usually constructed with the help of the Axiom of Choice are definable by $\Sigma_1$-formulas that only use the…

A simplified construction is presented for Komj\'ath's result that for every uncountable cardinal $\kappa$, there are $2^\kappa$ graphs of size $\kappa$ none of them being a minor of another.

Let $p/q$ ($p, q \in \mathbb{N}^*$) be a positive rational number such that $p > q^2$. We show that for any $\epsilon > 0$, there exists a set $A(\epsilon) \subset [0, 1[$, with finite border and with Lebesgue measure $< \epsilon$, for…

Assume G.C.H. and kappa is the first uncountable cardinal such that there is a kappa-free abelian group which is not a Whitehead (abelian) group. We prove that kappa is necessarily an inaccessible cardinal

Answering some of the main questions from [MR13], we show that whenever $\kappa$ is a cardinal satisfying $\kappa^{< \kappa} = \kappa > \omega$, then the embeddability relation between $\kappa$-sized structures is strongly invariantly…

For a commutative, unital and divisible quantale $\mathsf{Q}$, it is shown that the category of $\mathsf{Q}$-sets is a topos if, and only if, $\mathsf{Q}$ is a frame.

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(\kappa)$, for a regular cardinal $\kappa$, and ${\cal C}(V)$ (the class of…

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory,…

A MAD (maximal almost disjoint) family is an infinite subset A of the infinite subsets of {0,1,2,..} such that any two elements of A intersect in a finite set and every infinite subset of {0.1.2...} meets some element of $\aa$ in an…

We present an overview of results on the question of whether the non-stationary ideal of an uncountable regular cardinal $\kappa$ can be defined by a $\Pi_1$-formula using parameters of hereditary cardinality at most $\kappa$. These results…

This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by…

For an uncountable regular cardinal \kappa we let \nabla_\kappa(A) be the statement that A \subset \kappa and for all regular \theta > \kappa, the set of all X \in [\theta]^<\kappa such that X \cap \kappa \in \kappa and otp(X \cap OR) is a…

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $ of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies c.c.c.)…

We study the possible structures which can be carried by sets which have no countable subset, but which fail to be `surjectively Dedekind finite', in two possible senses, that there is a surjection to $\omega$, or alternatively, that there…

Cantor's famous proof of the non-denumerability of real numbers does apply to any infinite set. The set of exclusively all natural numbers does not exist. This shows that the concept of countability is not well defined. There remains no…

Techniques of combinatorial set theory are applied to the following algebraic problem. Suppose G is an abelian group such that, for all countable subgroups C, the divisible part of the quotient G/C is countable. What can one conclude about…