Coloring triangles and rectangles

Jindřich Zapletal

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 1, page 83-96
  • ISSN: 0010-2628

Abstract

top
It is consistent that ZF + DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles on ℝ 2 does not.

How to cite

top

Zapletal, Jindřich. "Coloring triangles and rectangles." Commentationes Mathematicae Universitatis Carolinae 64.1 (2023): 83-96. <http://eudml.org/doc/299364>.

@article{Zapletal2023,
abstract = {It is consistent that ZF + DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles on $\mathbb \{R\}^2$ does not.},
author = {Zapletal, Jindřich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {real algebraic geometry; algebraic hypergraph; chromatic number; geometric set theory; consistency result},
language = {eng},
number = {1},
pages = {83-96},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Coloring triangles and rectangles},
url = {http://eudml.org/doc/299364},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Zapletal, Jindřich
TI - Coloring triangles and rectangles
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 1
SP - 83
EP - 96
AB - It is consistent that ZF + DC holds, the hypergraph of rectangles on a given Euclidean space has countable chromatic number, while the hypergraph of equilateral triangles on $\mathbb {R}^2$ does not.
LA - eng
KW - real algebraic geometry; algebraic hypergraph; chromatic number; geometric set theory; consistency result
UR - http://eudml.org/doc/299364
ER -

References

top
  1. Ceder J., Finite subsets and countable decompositions of Euclidean spaces, Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251. MR0257307
  2. Erdös P., Kakutani S., 10.1090/S0002-9904-1943-07954-2, Bull. Amer. Math. Soc. 49 (1943), 457–461. MR0008136DOI10.1090/S0002-9904-1943-07954-2
  3. Erdös P., Komjáth P., Countable decompositions of ℝ 2 and ℝ 3 , Discrete Comput. Geom. 5 (1990), no. 4, 325–331. MR1043714
  4. Ihoda J. I., Shelah S., 10.2307/2274613, J. Symbolic Logic 53 (1998), no. 4, 1188–1207. MR0973109DOI10.2307/2274613
  5. Jech T., Set Theory, Springer Monographs in Mathematics, Springer, Berlin, 2003. Zbl1007.03002MR1940513
  6. Larson P., Zapletal J., 10.1090/surv/248, Mathematical Surveys and Monographs, 248, American Mathematical Society, Providence, 2020. MR4249448DOI10.1090/surv/248
  7. Marker D., Model Theory: An Introduction, Graduate Texts in Mathematics, 217, Springer, New York, 2002. MR1924282
  8. Schmerl J. H., 10.1090/S0002-9947-99-02331-4, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2479–2489. MR1608502DOI10.1090/S0002-9947-99-02331-4
  9. Zapletal J., Noetherian spaces in choiceless set theory, available at arXiv:2101.03434v3 [math.LO] (2022), 23 pages. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.