We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable. (3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...

We show that all sufficiently nice λ-sets are countable dense homogeneous (𝖢𝖣𝖧). From this fact we conclude that for every uncountable cardinal κ ≤ 𝔟 there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every 𝖢𝖣𝖧 metric space has size either ω₁ or 𝔠. An...

The following theorem is proved, answering a question raised by Davies in 1963. If $L_0\cup L_1\cup L_2\cup ...$ is a partition of the set of lines of ${ℝ}^n$, then there is a partition ${ℝ}^n=S_0\cup S_1\cup S_2\cup ...$ such that $|\ell \cap S_i|\le 2$ whenever $\ell \in L_i$. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson Mauldin.

A graph is called splitting if there is a 0-1 labelling of its vertices such that for every infinite set C of natural numbers there is a sequence of labels along a 1-way infinite path in the graph whose restriction to C is not eventually constant. We characterize the countable splitting graphs as those containing a subgraph of one of three simple types.

We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a $G_{\delta }$. In addition some nonperfect spaces with σ-disjoint bases are constructed.

Let T denote a completion of ZF. We are interested in the number μ(T) of isomorphism types of countable well-founded models of T. Given any countable order type τ, we are also interested in the number μ(T,τ) of isomorphism types of countable models of T whose ordinals have order type τ. We prove: (1) Suppose ZFC has an uncountable well-founded model and $\kappa \in \omega \cup \aleph ₀,\aleph ₁,2^{\aleph ₀}$. There is some completion T of ZF such that μ(T) = κ. (2) If α <ω₁ and μ(T,α) > ℵ₀, then $\mu (T,\alpha )=2^{\aleph ₀}$. (3) If α < ω₁ and T ⊢ V ≠ OD, then $\mu (T,\alpha )\in 0,2^{\aleph ₀}$. (4)...

We continue the work of Shelah and Casanovas on the cardinality of families of pairwise inconsistent types in simple theories. We prove that, in a simple theory, there are at most ${\lambda }^{<\kappa \left(T\right)}+2^{\mu +\left|T\right|}$ pairwise inconsistent types of size μ over a set of size λ. This bound improves the previous bounds and clarifies the role of κ(T). We also compute exactly the maximal cardinality of such families for countable, simple theories. The main tool is the fact that, in simple theories, the collection of nonforking extensions...

We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.

Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if $C_{EK}$ is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of $C_{EK}$. As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result holds for...