arXiv subjects

Václav Vlasák

Publications and source records attributed to Václav Vlasák.

7 recordsLinked to original sources

arXiv2026

A nonmicroscopic $G_δ$ set with only microscopic compact subsets

We construct a nonmicroscopic $G_δ$ subset of the real line all of whose compact subsets are microscopic, answering a question of Zindulka and Nowakowski. We first obtain such a set in the standard Cantor space by coding branches of a finitely branching tree together with suitable choices of marked nodes. The coding of the branches ensures that the resulting set is not microscopic, while the marked nodes allow us to construct covers showing that every compact subset is microscopic. The example is then transferred from the Cantor space to the real line by the binary-expansion map.

math.LO↗

arXiv2020

A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems

Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as well as the $d$-dimensional Lebesgue measure of the vertex set of a large-distance graph in the $d$-dimensional Euclidean space that contains no copies of a complete graph on $k$ vertices. The former problem may be seen as a continuous analogue of Turán's classical graph theorem, and the latter as a graph-theoretic analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest.

math.CO↗

arXiv2019

On $k$-antichains in the unit $n$-cube

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \[ \text{card}(A \cap C) \le k, \, \text{ for all chains } \, C \subset [0,1]^n \, , \] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. We show that the $(n-1)$-dimensional Hausdorff measure of a $k$-antichain in $[0,1]^n$ is at most $kn$ and that the bound is asymptotically sharp. Moreover, we conjecture that there exist $k$-antichains in $[0,1]^n$ whose $(n-1)$-dimensional Hausdorff measure equals $kn$ and we verify the validity of this conjecture when $n=2$.

math.CA↗

arXiv2019

A continuous analogue of Erdős' $k$-Sperner theorem

A \emph{chain} in the unit $n$-cube is a set $C\subset [0,1]^n$ such that for every $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in $C$ we either have $x_i\le y_i$ for all $i\in [n]$, or $x_i\ge y_i$ for all $i\in [n]$. We show that the $1$-dimensional Hausdorff measure of a chain in the unit $n$-cube is at most $n$, and that the bound is sharp. Given this result, we consider the problem of maximising the $n$-dimensional Lebesgue measure of a measurable set $A\subset [0,1]^n$ subject to the constraint that it satisfies $\mathcal{H}^1(A\cap C) \le κ$ for all chains $C\subset [0,1]^n$, where $κ$ is a fixed real number from the interval $(0,n]$. We show that the measure of $A$ is not larger than the measure of the following optimal set: \[ A^{\ast}_κ = \left\{ (x_1,\ldots,x_n)\in [0,1]^n : \frac{n-κ}{2}\le \sum_{i=1}^{n}x_i \le \frac{n+ κ}{2} \right\} \, . \] Our result may be seen as a continuous counterpart to a theorem of Erdős, regarding $k$-Sperner families of finite sets.

math.CA↗

arXiv2016

Haar meager sets, their hulls, and relationship to compact sets

Let $G$ be an abelian Polish group. We show that there is a strongly Haar meager set in $G$ without any $F_σ$ Haar meager hull (and that this still remains true if we replace $F_σ$ by any other class of the Borel hierarchy). We also prove that there is a coanalytic naively strongly Haar meager set without any Haar meager hull. Further, we investigate the relationship of the collection of all compact sets to the collection of all Haar meager sets in non-locally compact Polish groups.

math.GN↗

arXiv2015

Haar meager sets revisited

In the present article we investigate Darji's notion of Haar meager sets from several directions. We consider alternative definitions and show that some of them are equivalent to the original one, while others fail to produce interesting notions. We define Haar meager sets in nonabelian Polish groups and show that many results, including the facts that Haar meager sets are meager and form a $σ$-ideal, are valid in the more general setting as well. The article provides various examples distinguishing Haar meager sets from Haar null sets, including decomposition theorems for some subclasses of Polish groups. As a corollary we obtain, for example, that $\mathbb Z^ω$, $\mathbb R^ω$ or any Banach space can be decomposed into a Haar meager set and a Haar null set. We also establish the stability of non-Haar meagerness under Cartesian product.

math.GN↗

arXiv2012

Properties of functions with monotone graphs

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, e.g., that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that Hausdorff dimension of the graph of such a function is 1.

math.CA↗