arXiv subjects
Publications and source records attributed to Miroslav Zeleny.
We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of antichains of size continuum, and that there is no finite basis. 2010 Mathematics Subject Classification. 03E15, 54H05
The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G\_{δσ}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${\bfΠ}^1\_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${\bfΠ}^1\_1$ and ${\bfΣ}^1\_2$ in the theory of martingales.
We construct, for each countable ordinal $ξ$, a closed graph with Borel chromatic number two and Baire class $ξ$ chromatic number $\aleph\_0$.
We use a method involving elementary submodels and a partial converse of Foran lemma to prove separable reduction theorems concerning Suslin sigma-P-porous sets where "P" can be from a rather wide class of porosity-like relations in complete metric spaces. In particular, we separably reduce the notion of Suslin cone small set in Asplund spaces. As an application we prove a theorem stating that a continuous approximately convex function on an Asplund space is Frechet differentiable up to a cone small set.
We give, for each countable ordinal $ξ\geq 1$, an example of a ${\bfΔ}^0_2$ countable union of Borel rectangles that cannot be decomposed into countably many ${\bfΠ}^0_ξ$ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ${\bfΔ}^0_ξ$-measurable countable coloring.
The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\vcević characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb{G}_0$-dichotomy for $\boraxi$-measurable countable colorings when $ξ\leq 3$. A $\boraxi$-measurable countable coloring gives a covering of the diagonal consisting of countably many $\boraxi$ squares. This leads to the study of countable unions of $\boraxi$ rectangles. We also give a Hurewicz-like dichotomy for such countable unions when $ξ\leq 2$.