Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
real algebraic geometry; algebraic hypergraph; chromatic number; geometric set theory; consistency result
real algebraic geometry; algebraic hypergraph; chromatic number; geometric set theory; consistency result
Summary:
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.
References:
[1] Ceder J.: Finite subsets and countable decompositions of Euclidean spaces. Rev. Roumaine Math. Pures Appl. 14 (1969), 1247–1251. MR 0257307
[2] Erdös P., Kakutani S.: On non-denumerable graphs. Bull. Amer. Math. Soc. 49 (1943), 457–461. DOI 10.1090/S0002-9904-1943-07954-2 | MR 0008136
[3] Erdös P., Komjáth P.: Countable decompositions of $\mathbb{R}^2$ and $\mathbb{R}^3$. Discrete Comput. Geom. 5 (1990), no. 4, 325–331. MR 1043714
[4] Ihoda J. I., Shelah S.: Souslin forcing. J. Symbolic Logic 53 (1998), no. 4, 1188–1207. DOI 10.2307/2274613 | MR 0973109
[5] Jech T.: Set Theory. Springer Monographs in Mathematics, Springer, Berlin, 2003. MR 1940513 | Zbl 1007.03002
[6] Larson P., Zapletal J.: Geometric Set Theory. Mathematical Surveys and Monographs, 248, American Mathematical Society, Providence, 2020. DOI 10.1090/surv/248 | MR 4249448
[7] Marker D.: Model Theory: An Introduction. Graduate Texts in Mathematics, 217, Springer, New York, 2002. MR 1924282
[8] Schmerl J. H.: Avoidable algebraic subsets of Euclidean space. Trans. Amer. Math. Soc. 352 (2000), no. 6, 2479–2489. DOI 10.1090/S0002-9947-99-02331-4 | MR 1608502
[9] Zapletal J.: Noetherian spaces in choiceless set theory. available at arXiv:2101.03434v3 [math.LO] (2022), 23 pages.
Search
Partner of
