Related papers: Analytic digraphs of uncountable Borel chromatic n…

We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the…

We study several natural classes of graphs on a zero-dimensional metrizable compact space having no continuous coloring. We compare these graphs with the quasi-order associated with injective continuous homomorphisms. We prove the existence…

We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we…

I prove that the Borel directed graphs whose vertex set admits a partition into two Borel acyclic sets form a $\mathbf\Sigma^1_2$-complete set; equivalently, that deciding whether a Borel directed graph has Borel dichromatic number at…

In this work, we continue the tradition initiated by Geschke, 2011 of viewing the uncountable Borel chromatic number of analytic graphs as cardinal invariants of the continuum. We show that various uncountable Borel chromatic numbers of…

In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally…

We study the class of analytic binary relations on Polish spaces, compared with the notions of continuous reducibility or injective continuous reducibility. In particular, we characterize when a locally countable Borel relation is $\Sigma$…

We have observations concerning the set theoretic strength of the following combinatorial statements without the axiom of choice. 1. If in a partially ordered set, all chains are finite and all antichains are countable, then the set is…

We present an elementary construction of an uncountably chromatic graph without uncountable, infinitely connected subgraphs.

Motivated by an old conjecture of P. Erd\H{o}s and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable…

A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain for example $ C_4 $ (among other obligatory subgraphs). It was shown recently by D. T. Soukup that, in contrast of…

In this paper, we study definable variants of the notion of the distinguishing number of a graph in descriptive set theoretic setting. We introduce the notion of the Borel distinguishing number of a Borel graph and provide examples that…

Let $X$ be a Polish space with Borel probability measure $\mu,$ and let $G$ be a Borel graph on $X$ with no odd cycles and maximum degree $\Delta(G).$ We show that the Baire measurable edge chromatic number of $G$ is at most $\Delta(G)+1$,…

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions.…

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence…

We study the isomorphism relation on Borel classes of locally compact Polish metric structures. We prove that isomorphism on such classes is always classifiable by countable structures (equivalently: Borel reducible to graph isomorphism),…

We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on $[\mathbb{N}]^{<\mathbb{N}}$ with finite (or,…

The $\mathbb{G}_0$-dichotomy due to Kechris, Solecki and Todor\vcevi\'c characterizes the analytic relations having a Borel-measurable countable coloring. We give a version of the $\mathbb{G}_0$-dichotomy for $\boraxi$-measurable countable…

We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the…

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge…