arXiv subjects
Publications and source records attributed to Jan Grebik.
Extending a result of Christiansen, we prove that every mutli-graph $G=(V,E)$ admits a proper edge colouring $\phi:E\to \{1,2,\dots\}$ which is local, that is, $\phi(e)\le \max\{d(x)+\pi(x),d(y)+\pi(y)\}$ for every edge $e$ with end-points $x,y\in V$, where $d(z)$ (resp.\ $\pi(z)$) denotes the degree of a vertex $z$ (resp.\ the maximum edge multiplicity at $z$). This is derived from a local version of the Fan Equation.
Following recent result of L. M. T\' oth [arXiv:1906.03137] we show that every $2Δ$-regular Borel graph $\mathcal{G}$ with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig's line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.
We show that every essentially countable orbit equivalence relation induced by a continuous action of a Polish group on a Polish space is $σ$-lacunary. In combination with [Invent. Math.201 (1), 309-383, 2015] we obtain a straightforward proof of the result from [Adv. Math.307, 312-343,2017] that every essentially countable equivalence relation that is induced by an action of abelian non-archimedean Polish group is essentially hyperfinite.
We show that there is a Borel way of choosing a representative of a $k$-uniform hypergraphon. This extends the result of Orbanz and Szegedy where this was shown for graphons.
We show that the Nash-Williams theorem has a uniform version and that the Galvin theorem does not. We show that there is an $F_σ$ tall ideal on $\mathbb{N}$ without a Borel selector and also construct a $\mathbfΠ^1_2$ tall ideal without a tall closed subset.
A topological space $X$ is said to be an Ascoli space if any compact subset $K$ of $C_k(X)$ is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the $k_R$-property and the Ascoli property of $C_p(κ)$ and $C_k(κ)$ over ordinals $κ$. We prove that $C_p(κ)$ is always an Ascoli space, while $C_p(κ)$ is a $k_R$-space iff the cofinality of $κ$ is countable. In particular, this provides the first $C_p$-example of an Ascoli space which is not a $k_R$-space, namely $C_p(ω_1)$. We show that $C_k(κ)$ is Ascoli iff $cf(κ)$ is countable iff $C_k(κ)$ is metrizable.