# Two dichotomy theorems on colourability of non-analytic graphs

Fundamenta Mathematicae (1997)

• Volume: 154, Issue: 2, page 183-201
• ISSN: 0016-2736

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## Abstract

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We prove:  Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph ${G}_{0}$ into G.  Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.

## How to cite

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Kanovei, Vladimir. "Two dichotomy theorems on colourability of non-analytic graphs." Fundamenta Mathematicae 154.2 (1997): 183-201. <http://eudml.org/doc/212233>.

@article{Kanovei1997,
abstract = {We prove:  Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph $G_0$ into G.  Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.},
journal = {Fundamenta Mathematicae},
keywords = {graph on reals; acyclic graph; locally thin graph; Borel colouring; definable colouring; dichotomies},
language = {eng},
number = {2},
pages = {183-201},
title = {Two dichotomy theorems on colourability of non-analytic graphs},
url = {http://eudml.org/doc/212233},
volume = {154},
year = {1997},
}

TY - JOUR
TI - Two dichotomy theorems on colourability of non-analytic graphs
JO - Fundamenta Mathematicae
PY - 1997
VL - 154
IS - 2
SP - 183
EP - 201
AB - We prove:  Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a κ-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph $G_0$ into G.  Theorem 2. In the Solovay model, every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above.
LA - eng
KW - graph on reals; acyclic graph; locally thin graph; Borel colouring; definable colouring; dichotomies
UR - http://eudml.org/doc/212233
ER -

## References

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1. [1] D. Guaspari, Trees, norms, and scales, in: London Math. Soc. Lecture Note Ser. 87, Cambridge Univ. Press, 1983, 135-161. Zbl0549.03040
2. [2] L. A. Harrington, A. S. Kechris and A. Louveau, A Glimm-Effros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903-928. Zbl0778.28011
3. [3] L. A. Harrington and S. Shelah, Counting equivalence classes for co-κ-Souslin equivalence relations, in: D. van Dalen et al. (eds.), Logic Colloquium '80 (Prague, 1980), North-Holland, 1982, 147-152.
4. [4] G. Hjorth, Thin equivalence relations and effective decompositions, J. Symbolic Logic 58 (1993), 1153-1164. Zbl0793.03051
5. [5] G. Hjorth, A remark on ${\prod }_{1}^{1}$ equivalence relations, handwritten note.
6. [6] V. Kanovei, An Ulm-type classification theorem for equivalence relations in Solovay model, J. Symbolic Logic, to appear. Zbl0895.03020
7. [7] V. Kanovei, On a dichotomy related to colourings of definable graphs in generic models, preprint ML-96-10, University of Amsterdam, 1996.
8. [8] A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.
9. [9] A. S. Kechris, S. Solecki and S. Todorčević, Borel chromatic numbers, Adv. Math., to appear. Zbl0100.24105
10. [10] R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. 92 (1970), 1-56. Zbl0207.00905

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