arXiv subjects

Tamás Mátrai

Publications and source records attributed to Tamás Mátrai.

4 recordsLinked to original sources

arXiv2013

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal P(ω)$ embeds into the Borel Tukey ordering on cardinal invariants.

math.LO↗

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↗

arXiv2009

On splitting infinite-fold covers

Let $X$ be a set, $\ka$ be a cardinal number and let $\iH$ be a family of subsets of $X$ which covers each $x\in X$ at least $\ka$ times. What assumptions can ensure that $\iH$ can be decomposed into $κ$ many disjoint subcovers? We examine this problem under various assumptions on the set $X$ and on the cover $\iH$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of $\real^{n}$ by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite $κ$. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

math.CO↗

arXiv2009

On $\ell^{p}$-like equivalence relations

For $f \colon [0,1] \rar \real^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^ω$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n < ω} f(|y_{n} - x_{n}|) < \infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\mathbf{E}_{f}$. Our results indicate that for every $1 \leq p < q < \infty$, the order $\leq_{B}$ of Borel reducibility on the set of equivalence relations $\{\bE \colon \bE_{\Id^{p}} \leq_{B} \bE \leq_{B} \bE_{\Id^{q}}\}$ is more complicated than expected, e.g. consistently every linear order of cardinality continuum embeds into it.

math.LO↗