scholarly journals Minimal definable graphs of definable chromatic number at least three

2021 ◽  
Vol 9 ◽  
Author(s):  
Raphaël Carroy ◽  
Benjamin D. Miller ◽  
David Schrittesser ◽  
Zoltán Vidnyánszky
Keyword(s):  
Chromatic Number ◽  
Analogous Result ◽  
Vertex Degree ◽  
Borel Space ◽  
Standard Borel Space

Abstract 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.

Bi-Borel reducibility of essentially countable Borel equivalence relations

2005 ◽  
Vol 70 (3) ◽  
pp. 979-992 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space ◽  
Borel Reducibility ◽  
New Feature ◽  
Borel Probability ◽  
Image Position

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

scholarly journals Amenable versus hyperfinite Borel equivalence relations

1993 ◽  
Vol 58 (3) ◽  
pp. 894-907 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Countable Group ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

scholarly journals Coboundaries of commuting Borel automorphisms

2021 ◽  
Vol 41 (5) ◽  
pp. 667-683
Author(s):  
Shrey Sanadhya
Keyword(s):  
Borel Space ◽  
Standard Borel Space

We show that if \(S\), \(T\) are two commuting automorphisms of a standard Borel space such that they generate a free Borel \(\mathbb{Z}^2\)-action then \(S\) and \(T\) do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel \(\mathbb{Z}^d\)-actions.

scholarly journals BROOKS’ THEOREM FOR MEASURABLE COLORINGS

2016 ◽  
Vol 4 ◽  
Author(s):  
CLINTON T. CONLEY ◽  
ANDREW S. MARKS ◽  
ROBIN D. TUCKER-DROB
Keyword(s):  
Probability Measure ◽  
Cayley Graphs ◽  
Borel Probability Measure ◽  
Group Actions ◽  
New Technique ◽  
Borel Space ◽  
Standard Borel Space ◽  
List Colorings ◽  
A New Technique ◽  
Borel Probability

We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$-cliques, then $G$ admits a ${\it\mu}$-measurable $d$-coloring with respect to any Borel probability measure ${\it\mu}$ 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 IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.

scholarly journals Colouring random geometric graphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Colin J. H. McDiarmid ◽  
Tobias Müller
Keyword(s):  
Chromatic Number ◽  
Analogous Result ◽  
Clique Number ◽  
Geometric Graph ◽  
Geometric Graphs ◽  
Random Geometric Graph ◽  
Random Geometric Graphs ◽  
Different Types ◽  
International Audience ◽  
Uniform Case

International audience A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (03) ◽  
pp. 567-581 ◽  
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Independence Number ◽  
Rates Of Convergence ◽  
Clique Number ◽  
Vertex Degree ◽  
Edge Distance

On independent random points U 1 ,· ··,Un distributed uniformly on [0, 1] d , a random graph Gn (x) is constructed in which two distinct such points are joined by an edge if the l ∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn } are provided which guarantee the random graph to be empty of edges, a.s.

On Polish groups admitting a compatible complete left-invariant metric

2011 ◽  
Vol 76 (2) ◽  
pp. 437-447 ◽  
Author(s):  
Maciej Malicki
Keyword(s):  
Invariant Metric ◽  
Borel Space ◽  
Polish Groups ◽  
Standard Borel Space

AbstractWe prove that the set of all Polish groups admitting a compatible complete left-invariant metric (called CLI) is coanalytic non-Borel as a subset of a standard Borel space of all Polish groups. As an application of this result, we show that there does not exist a weakly universal CLI group. This, in particular, answers in the negative a question of H.Becker.

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

1997 ◽  
Vol 29 (3) ◽  
pp. 567-581 ◽  
Author(s):  
Martin J. B. Appel ◽  
Ralph P. Russo
Keyword(s):  
Chromatic Number ◽  
Random Graph ◽  
Independence Number ◽  
Rates Of Convergence ◽  
Clique Number ◽  
Vertex Degree ◽  
Edge Distance

On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.

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