We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by $k$ for various cardinals $k$. We show that for a p.m.p. graph $G$, a measurable orientation can be found when $k$ is larger than the normalized cost of the restriction of $G$ to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every $k$ we also find graphs which admit measurable orientations with outdegree bounded by $k$ but no such Borel orientations. Finally, for special values of $k$ we bound the projective complexity of Borel $k$-orientability for graphs and graphings of equivalence relations. It follows from these bounds that the set of equivalence relations admitting a Borel selector is $\mathbfΣ^1_2$ in the codes, in stark contrast to the case of smooth relations.
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 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 characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.
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 closed graphs can be consistently different, as well as consistently equal to the continuum. This is done using arguments that are typical to Axiom A forcing notions.
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 connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, local rules, Borel transversals, etc. Also, we present the construction of Andrew Marks of acyclic Borel graphs for which …
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 the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a c…
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 disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.
We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a non-separation result for iterated Frechet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, …
In this paper, we show that Frucht's theorem holds in Borel setting. More specifically, we prove that any standard Borel group can be realized as the Borel automorphism group of a Borel graph. A slight modification of our construction also yields the following result in topological setting: Any Polish group can be realized as the homeomorphic automorphism group of a $\mathbf{Δ^0_2}$-graph on a Polish space.
Let $Γ$ be a countable group and let $G$ be the Schreier graph of the free part of the Bernoulli shift of $Γ$ (with respect to some finite subset $F \subseteq Γ$). We show that the Borel fractional chromatic number of $G$ is equal to $1$ over the measurable independence number of $G$. As a consequence, we asymptotically determine the Borel fractional chromatic number of $G$ when $Γ$ is the free group, answering a question of Meehan.
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, equivalently, $\leq 3$) Borel chromatic number form a $\mathbfΣ^1_2$-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.
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 separate distinguishing number and Borel distinguishing number at various levels. More specifically, we prove that there exist Borel graphs with countable distinguishing number but uncountable Borel distinguishing number and that, for every integer $n \geq 3$, there exists a Borel graph with distinguishing number $2$ whose Borel distinguishing number is finite …
We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are "far" from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.
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 Baire measurable and measure chromatic numbers, answering analogous versions of the question.
We construct a Borel graph G such that ZF+DC+"There are no maximal independent sets in G" is equiconsistent with ZFC+"There exists an inaccessible cardinal".
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 chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d$-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also gi…
Hajnal and Szemerédi proved that if $G$ is a finite graph with maximum degree $Δ$, then for every integer $k \geqslant Δ+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by $1$; such colorings are called equitable. We obtain an analog of this result for infinite graphs in the Borel setting. Specifically, we show that if $G$ is an aperiodic Borel graph of finite maximum degree $Δ$, then for each $k \geqslant Δ+ 1$, $G$ has a Borel proper $k$-coloring in which every two color classes are related by an element of the Borel full semigroup of $G$…
We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $μ$-measurable $d$-coloring with respect to any Borel probability measure $μ$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of …
Let $(X,τ)$ be a Polish space with Borel probability measure $μ,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of an amenable (resp., polynomial growth) group then we show the same result $μ$-a.e. (resp., everywhere). Our results generalize recent work of Timár, as well as of Conley, Gaboriau, Marks, and Tucker-Drob, who proved this in the probability measure preserving setting. We apply our theorem to find Borel orientations in even degree graphs and measurable and Baire mea…
We show that Vizing's Theorem holds in the Borel context for graphs induced by actions of 2-ended groups, and ask whether it holds more generally for everywhere two ended Borel graphs.
We construct a smooth locally finite Borel graph $G$ and a local coloring problem $Π$ such that $G$ has a coloring $V(G) \to \mathbb{N}$ that solves $Π$, but no such coloring can be Borel.
We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and "Can marked groups with isomorphic Cayley graphs have Borel chromatic numbers for their shift graphs which differ by more than one?" We also provide a new bound for Borel chromatic numbers of graphs whose connected components all have two ends.
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 explicitly constructed. In this paper, we show that examples cannot be constructed via standard methods. Moreover, we show that most of the known better quasi-orders are non-examples, suggesting there is still a class of better quasi-orders with interesting combinatorial properties who's elements/members still remain unknown.
Borel completeness of the class of countable Steiner triple systems
School of Mathematics\ University 24 South Section 1, First Ring Road\ , Sichuan 610065 China；School of Mathematics；Statistics\ Jiaotong University 3 Shangyuancun\ , Beijing 100044 China
We prove that consistently, cov($\mathcal{M})< λ_\mathbf{0} < λ_\mathbf{1} < λ_\mathbf{\infty} < 2^{\aleph_0}$, where $λ_\mathbf{0}$ denotes the weak Borel chromatic number of the Kechris-Solecki-Todorčević graph $\mathbb{G}_0$, that is, the minimal cardinality of a $\mathbb{G}_0$-independent Borel covering of $2^ω$, while $λ_\mathbf{1}$ and $λ_\infty$ are the corresponding invariants of the graph $\mathbb{G}_1$ and the simple graph associated with the equivalence relation $E_0$.
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 for every ordinal $α\in [1, ω_1)$ there is a closed set $F \subset 2^ω\times ω^ω$ such that for every $x \in 2^ω$ the section $\{y\in ω^ω; (x,y) \in F\}$ is a two-point set and $F$ cannot be covered by countably many graphs $B(n) \subset 2^ω\times ω^ω$ of functions of the variable $x \in 2^ω$ such that each $B(n)$ is in the additive Borel class $\boldsymbol Σ^0_α$. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable $Π^0_1$ se…
