Measurable chromatic numbers

2008 ◽  
Vol 73 (4) ◽  
pp. 1139-1157 ◽  
Author(s):  
Benjamin D. Miller
Keyword(s):  
Chromatic Number ◽  
Equivalence Relation ◽  
Initial Segment ◽  
Polish Space ◽  
Borel Function ◽  
Chromatic Numbers ◽  
Unilateral Shift ◽  
Dichotomy Theorem ◽  
Borel Functions ◽  
Measurable Chromatic Number

AbstractWe show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [ℕ]ℕ, although its Borel chromatic number is ℵ0. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for “the” minimum analytic graph with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all κ Є6 (2, 3…..ℵ0, c), there is a treeing of E0 with Borel and Baire measurable chromatic number κ. Finally, we use a Glimm–Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (ℝ<ℕ, ⊇).

A dichotomy theorem for turbulence

2002 ◽  
Vol 67 (4) ◽  
pp. 1520-1540 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Borel Function ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Borel Set ◽  
Polish Group ◽  
Strengthened Version ◽  
Dichotomy Theorem ◽  
Image Position

In this note we show:Theorem 1.1. Let G be a Polish group and X a Polish G-space with the induced orbit equivalence relation EG Borel as a subset of X × X. Then exactly one of the following:(I) There is a countable languageℒand a Borel functionsuch that for all x1, x2 ∈ Xor(II) there is a turbulent Polish G-space Y and a continuous G-embeddingThere are various bows and ribbons which can be woven into these statements. We can strengthen (I) by asking that θ also admit a Borel orbit inverse, that is to say some Borel functionfor some Borel set B ⊂ Mod(ℒ), such that for all x ∈ Xand then after having passed to this strengthened version of (I) we still obtain the exact same dichotomy theorem, and hence the conclusion that the two competing versions of (I) are equivalent. Similarly (II) can be relaxed to just asking that τ be a Borel G-embedding, or even simply a Borel reduction of the relevant orbit equivalence relations. It is in fact a consequence of 1.1 that all the plausible weakenings and strengthenings of (I) and (II) are respectively equivalent to one another.I will not closely examine these possible variations here. The equivalences alluded to above follow from our main theorem and the results of [3]. That monograph had previously shown that (I) and (II) are incompatible, and proved a barbaric forerunner of 1.1, and gone on to conjecture the dichotomy result above.

scholarly journals Analytic equivalence relations and Ulm-type classifications

1995 ◽  
Vol 60 (4) ◽  
pp. 1273-1300 ◽  
Author(s):  
Greg Hjorth ◽  
Alexander S. Kechris
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Borel Measure ◽  
Polish Space ◽  
Borel Function ◽  
Equivalence Relations ◽  
Cantor Space ◽  
Continuous Action ◽  
Borel Set ◽  
Continuous Embedding

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.The original Glimm-Effros dichotomy, established by Effros [Ef], [Ef1], who generalized work of Glimm [G1], asserts that if an Fσ equivalence relation on a Polish space X is induced by the continuous action of a Polish group G on X, then exactly one of the following alternatives holds:(I) Elements of X can be classified up to E-equivalence by “concrete invariants” computable in a reasonably definable way, i.e., there is a Borel function f: X → Y, Y a Polish space, such that xEy ⇔ f(x) = f(y), or else(II) E contains a copy of a canonical equivalence relation which fails to have such a classification, namely the relation xE0y ⇔ ∃n∀m ≥ n(x(n) = y(n)) on the Cantor space 2ω (ω = {0,1,2, …}), i.e., there is a continuous embedding g: 2ω → X such that xE0y ⇔ g(x)Eg(y).Moreover, alternative (II) is equivalent to:(II)′ There exists an E-ergodic, nonatomic probability Borel measure on X, where E-ergodic means that every E-invariant Borel set has measure 0 or 1 and E-nonatomic means that every E-equivalence class has measure 0.

On the measurable chromatic number of a space of dimension n ≤ 24

2015 ◽  
Vol 92 (3) ◽  
pp. 761-763 ◽  
Author(s):  
L. I. Bogolubsky ◽  
A. M. Raigorodskii
Keyword(s):  
Chromatic Number ◽  
Measurable Chromatic Number

scholarly journals Bounds on the measurable chromatic number of Rn

1989 ◽  
Vol 75 (1-3) ◽  
pp. 343-372 ◽  
Author(s):  
L.A. Székely ◽  
N.C. Wormald
Keyword(s):  
Chromatic Number ◽  
Measurable Chromatic Number

On the Babai and Upper Chromatic Numbers of Graphs of Diameter 2

2021 ◽  
Vol 52 (1) ◽  
pp. 113-123
Author(s):  
Peter Johnson ◽  
Alexis Krumpelman
Keyword(s):  
Chromatic Number ◽  
Metric Space ◽  
Simple Graph ◽  
Chromatic Numbers

The Babai numbers and the upper chromatic number are parameters that can be assigned to any metric space. They can, therefore, be assigned to any connected simple graph. In this paper we make progress in the theory of the first Babai number and the upper chromatic number in the simple graph setting, with emphasis on graphs of diameter 2.

A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

2006 ◽  
Vol 71 (4) ◽  
pp. 1108-1124 ◽  
Author(s):  
Alex Thompson
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Invariant Metrics ◽  
Orbit Equivalence ◽  
Invariant Metric ◽  
Continuous Action ◽  
Polish Groups ◽  
Polish Group ◽  
Left Invariant Metrics

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

2013 ◽  
Vol 333-335 ◽  
pp. 1452-1455
Author(s):  
Chun Yan Ma ◽  
Xiang En Chen ◽  
Fang Yang ◽  
Bing Yao
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Adjacent Vertex ◽  
Chromatic Numbers ◽  
Edge Colorings ◽  
Proper Edge Coloring ◽  
Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

PARALLEL VERTEX COLOURING OF INTERVAL GRAPHS

1999 ◽  
Vol 10 (01) ◽  
pp. 19-31 ◽  
Author(s):  
G. SAJITH ◽  
SANJEEV SAXENA
Keyword(s):  
Chromatic Number ◽  
Interval Graph ◽  
Interval Graphs ◽  
Chromatic Numbers ◽  
Erew Pram ◽  
Pram Model ◽  
Interval Representation ◽  
Left And Right ◽  
Crcw Pram ◽  
Vertex Colouring

Evidence is given to suggest that minimally vertex colouring an interval graph may not be in NC 1. This is done by showing that 3-colouring a linked list is NC 1-reducible to minimally colouring an interval graph. However, it is shown that an interval graph with a known interval representation and an O(1) chromatic number can be minimally coloured in NC 1. For the CRCW PRAM model, an o( log n) time, polynomial processors algorithm is obtained for minimally colouring an interval graph with o( log n) chromatic number and a known interval representation. In particular, when the chromatic number is O(( log n)1-ε), 0<ε<1, the algorithm runs in O( log n/ log log n) time. Also, an O( log n) time, O(n) cost, EREW PRAM algorithm is found for interval graphs of arbitrary chromatic numbers. The following lower bound result is also obtained: even when the left and right endpoints of the interval are separately sorted, minimally colouring an interval graph needs Ω( log n/ log log n) time, on a CRCW PRAM, with a polynomial number of processors.

Sign in / Sign up

Export Citation Format

Share Document