Related papers: Some partition properties for measurable colouring…

It is proved that there exists an (omega-1,omega-1) Souslin gap in the Boolean algebra (L(nu)/Fin,subseteq^*_ae) for every nonseparable measure nu. Thus a Souslin, also known as destructible, (omega-1,omega-1) gap in P(N)/Fin can always be…

We study the existence of non-separable compact spaces that support a measure and are small from the topological point of view. In particular, we show that under Martin's axiom there is a non-separable compact space supporting a measure…

In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem'. We explore the topological constraints on the existence of a (relaxed) measurable coloring of R^d…

We study partition properties for uncountable regular cardinals that arise by restricting partition properties defining large cardinal notions to classes of simply definable colourings. We show that both large cardinal assumptions and…

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability…

Given an action of a group $\Gamma$ on a measure space $\Omega$, we provide a sufficient criterion under which two sets $A, B\subseteq \Omega$ are measurably equidecomposable, i.e., $A$ can be partitioned into finitely many measurable…

We present several $\mathsf{ZFC}$ examples of compactifications $\gamma\omega$ of $\omega$ such that their remainders $\gamma\omega\backslash\omega$ are nonseparable and carry strictly positive measures.

For positive integers $k < n$ such that $k$ divides $n$, let $(n)^k_{\hom}$ be the set of homogeneous $k$-partitions of $\{1, \dots, n\}$, that is, the set of partitions of $\{1, \dots, n\}$ into $k$ classes of the same cardinality. In the…

We prove that there exists uncountably many pairwise disjoint open subsets of the Gelfand space of the measure algebra on any locally compact non-discrete abelian group which shows that this space is not separable (in fact, we prove this…

Assuming $\mathfrak b = \mathfrak c$ (or some weaker statement), we construct a compactification $\gamma\omega$ of $\omega$ such that its remainder $\gamma\omega\setminus\omega$ is nonseparable and carries a strictly positive measure.

We prove a full measurable version of Vizing's theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree uniformly bounded by $\Delta\in \mathbb{N}$ defined on a standard probability space…

We examine measure-theoretic properties of spaces constructed using certain technique of Todor\v{c}evi\'{c}. We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of…

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^2$, is a partition of $\mathbb{S}^2$ into parts such that no part contains a pair of orthogonal points, that is, a pair of points at spherical distance $\pi/2$ apart. It…

Holmsen, Kyn\v{c}l and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset…

We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which…

It is consistent for every (1 <= n< omega) that (2^omega = omega_n) and there is a function (F:[omega_n]^{< omega}-> omega) such that every finite set can be written at most (2^n-1) ways as the union of two distinct monocolored sets. If GCH…

We study measures on compact spaces by analyzing the properties of fibers of continuous mappings into 2^omega. We show that if a compact zerodimensional space K carries a measure of uncountable Maharam type, then such a mapping has a…

We study some non-local functionals on the Sobolev space $W^{1,p}_0(\Omega)$ involving a double integral on $\Omega\times\Omega$ with respect to a measure $\mu$. We introduce a suitable notion of convergence of measures on product spaces…

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra…

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any…