相关论文: Descriptive Chromatic Numbers of Locally Finite an…
We construct pairs of marked groups with isomorphic Cayley graphs but different Borel chromatic numbers for the free parts of their shift graphs. This answers a question of Kechris and Marks. We also show that these graphs have different…
We construct, for each countable ordinal $\xi$, a closed graph with Borel chromatic number two and Baire class $\xi$ chromatic number $\aleph\_0$.
In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or…
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep…
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 introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for…
We show that every $d$-regular bipartite Borel graph admits a Baire measurable $k$-regular spanning subgraph if and only if $d$ is odd or $k$ is even. This gives the first example of a locally checkable coloring problem which is known to…
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields…
In this work, we try to enunciate the Total chromatic number of some Cayley graphs like the Cayley graph on Symmetric group, Alternating group, Dihedral group with respect to some generating sets and some other regular graphs.
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,…
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…
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 $\Delta(G)+1$,…
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…
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…
Every better quasi-order codifies a Borel graph that does not contain a copy of the shift graph. It is known that there is a better quasi-order that codes a Borel graph with infinite Borel chromatic number, though one has yet to be…
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…
In this paper, we will study the chromatic number of Cayley graphs of algebraic groups that arise from algebraic constructions. Using Lang-Weil bound and representation theory of finite simple groups of Lie type, we will establish lower…
A locally checkable labeling problem (LCL) on a group $\Gamma$ asks one to find a labeling of the Cayley graph of $\Gamma$ satisfying a fixed, finite set of "local" constraints. Typical examples include proper coloring and perfect matching…
The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic…