Coloring the distance graphs. (English) Zbl 07722176
Summary: Let \(n\geqslant 1\) be a number. Let \(\Gamma_n\) be the graph on \(\mathbb{R}^n\) connecting points of rational Euclidean distance. It is consistent with choiceless set theory \(\mathrm{ZF} + \mathrm{DC}\) that \(\Gamma_n\) has countable chromatic number, yet the chromatic number of \(\Gamma_{n+1}\) is uncountable.
MSC:
| 03E35 | Consistency and independence results |
| 14P99 | Real algebraic and real-analytic geometry |
| 05C15 | Coloring of graphs and hypergraphs |
References:
| [1] | Bochnak, J.; Coste, M.; Roy, M-F, Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (1998), Berlin: Springer, Berlin · Zbl 0912.14023 · doi:10.1007/978-3-662-03718-8 |
| [2] | Erdős, P.; Hajnal, A., On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar., 17, 61-99 (1966) · Zbl 0151.33701 · doi:10.1007/BF02020444 |
| [3] | Erdős, P.; Komjáth, P., Countable decompositions of \(\mathbb{R}^2\) and \(\mathbb{R}^3\), Discrete Comput. Geom., 5, 4, 325-331 (1990) · Zbl 0723.52005 · doi:10.1007/BF02187793 |
| [4] | Jech, T., Set Theory Springer. Monographs in Mathematics (2003), Berlin: Springer, Berlin · Zbl 1007.03002 |
| [5] | Komjáth, P., A decomposition theorem for \(\mathbb{R}^n\), Proc. Amer. Math. Soc., 120, 3, 921-927 (1994) · Zbl 0805.04002 |
| [6] | Larson, PB; Zapletal, J., Geometric Set Theory. Mathematical Surveys and Monographs (2020), Providence: American Mathematical Society, Providence · Zbl 07269808 · doi:10.1090/surv/248 |
| [7] | Marker, D., Model Theory: An Introduction. Graduate Texts in Mathematics (2002), New York: Springer, New York · Zbl 1003.03034 |
| [8] | Schmerl, JH, Avoidable algebraic subsets of Euclidean space, Trans. Amer. Math. Soc., 352, 6, 2479-2489 (2000) · Zbl 0948.03045 · doi:10.1090/S0002-9947-99-02331-4 |
| [9] | Shelah, S., Can you take Solovay’s inaccessible away?, Israel J. Math., 48, 1, 1-47 (1984) · Zbl 0596.03055 · doi:10.1007/BF02760522 |
| [10] | Zapletal, J.: Coloring the distance graphs in three dimensions (2021). arXiv:2103.02757 |
| [11] | Zapletal, J., Krull dimension in set theory, Ann. Pure Appl. Logic, 174, 9 (2023) · Zbl 07719409 · doi:10.1016/j.apal.2023.103299 |
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