A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/$G_{\delta }$ hull to every negligible/measurable subset of [0,1]? Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone $G_{\delta }$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent....

We calculate the cardinal characteristics of the $\sigma$-ideal $\mathscr{H}𝒩\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}𝒩\right(G\left)\right)\le 𝔟\le max\{𝔡,non(𝒩\left)\right\}\le non\left(\mathscr{H}𝒩\right(G\left)\right)\le cof\left(\mathscr{H}𝒩\right(G\left)\right)>min\{𝔡,non(𝒩\left)\right\}$. If $G={\prod }_{n\ge 0}{G}_n$ is the product of abelian locally compact groups $G_n$, then $add\left(\mathscr{H}𝒩\right(G\left)\right)=add\left(𝒩\right)$, $cov\left(\mathscr{H}𝒩\right(G\left)\right)=min\{𝔟,cov(𝒩\left)\right\}$, $non\left(\mathscr{H}𝒩\right(G\left)\right)=max\{𝔡,non(𝒩\left)\right\}$ and $cof\left(\mathscr{H}𝒩\right(G\left)\right)\ge cof\left(𝒩\right)$, where $𝒩$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}𝒩\left(G\right)\right)>2^{{\aleph }_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of...

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X² without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their homogeneity...

We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.

We study combinatorial principles we call the Homogeneity Principle HP(κ) and the Injectivity Principle IP(κ,λ) for regular κ > ℵ₁ and λ ≤ κ which are formulated in terms of coloring the ordinals < κ by reals. These principles are strengthenings of $C^s\left(\kappa \right)$ and $F^s\left(\kappa \right)$ of I. Juhász, L. Soukup and Z. Szentmiklóssy. Generalizing their results, we show e.g. that IP(ℵ₂,ℵ₁) (hence also IP(ℵ₂,ℵ₂) as well as HP(ℵ₂)) holds in a generic extension of a model of CH by Cohen forcing, and IP(ℵ₂,ℵ₂) (hence also HP(ℵ₂))...

We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$, ${}_{\mathbb{Q}}$ describing properties of Dense(ℚ). These invariants satisfy ${}_{\mathbb{Q}}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$.WecomparethemwiththeiranaloguesinthewellstudiedBooleanalgebra\left(\omega \right)/fin.Weshowthat$ℚ = p$,$ℚ = t$and$ℚ = i$,whereas$ℚ > h$and$ℚ > r$arebothshowntoberelativelyconsistentwithZFC.Wealsoinvestigatecombinatoricsoftheidealnwdofnowheredensesubsetsof,\mathbb{Q}.Inparticular,weshowthat$non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.

We prove that if there is a dominating family of size ℵ₁, then there are ℵ₁ many compact subsets of ${\omega }^{\omega }$ whose union is a maximal almost disjoint family of functions that is also maximal with respect to infinite partial functions.

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.

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...

We prove that it is consistent that the covering number of the ideal of measure zero sets has countable cofinality.

We formulate a Covering Property Axiom $CP{A}_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨fₙ\in {ℝ}^{ℝ}{⟩}_{n<\omega }$ of Borel functions there are sequences:...

We construct a compact set C of Hausdorff dimension zero such that cof(𝒩) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero-dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.