Abstract
In this work, the previous lower bound is considerably strengthened for the chromatic number of the nine-dimensional space.
Similar content being viewed by others
References
P. Brass, W. Moser, and J. Pach, Research Problems in Discrete Geometry, Springer, Berlin (2005).
K. Cantwell, “Finite Euclidean Ramsey theory,” J. Combin. Theory, Ser. A, 73, No. 2, 273–285 (1996).
J. Cibulka, “On the chromatic number of real and rational spaces,” Geombinatorics, 18, No. 2, 53–66 (2008).
A. E. Guterman, V. K. Lyubimov, A. M. Raigorodskii, and S. A. Usachev, “On the independence numbers of some distance graphs with vertices in {−1, 0, 1}n: estimates, conjectures, and applications to the Nelson–Erdős–Hadwiger and Borsuk problems,” Mat. Zametki (2009).
L. L. Ivanov, “An estimate for the chromatic number of the space ℝ4,” Usp. Mat. Nauk, 61, No. 5, 371–372 (2006).
N. N. Kuzyurin, “Asymptotic investigation of the set covering problem,” Probl. Kibern., No. 37, 19–56 (1980).
D. G. Larman and C. A. Rogers, “The realization of distances within sets in Euclidean space,” Mathematika, 19, 1–24 (1972).
L. Moser and W. Moser, “Solution to problem 10,” Can. Math. Bull., 4, 187–189 (1961).
O. Nechushtan, “On the space chromatic number,” Discrete Math., 256, 499–507 (2002).
A. M. Raigorodskii, “Systems of common representatives,” Fundam. Prikl. Mat., 5, No. 3, 851–860 (1999).
A. M. Raigorodskii, “The Borsuk problem and the chromatic numbers of some metric spaces,” Russ. Math. Surv., 56, No. 1, 103–139 (2001).
A. M. Raigorodskii, The Chromatic Numbers [in Russian], Moscow Center for Continuous Mathematical Education (MCCME), Moscow (2003).
A. M. Raigorodskii, The Linear Algebra Method in Combinatorics [in Russian], Moscow Center for Continuous Mathematical Education (MCCME), Moscow (2007).
A. Soifer, “The chromatic number of the plane: its past, present and future,” Mat. Prosveshchenie, No. 8 (2004).
L. A. Székely, “Erdős on unit distances and the Szemerédi–Trotter theorems,” in: G. Halász, ed., Paul Erdős and His Mathematics II. Based on Conf., Budapest, Hungary, July 4–11, 1999, Bolyai Soc. Math. Stud., Vol. 11, Springer, Berlin (2002), pp. 649–666.
V. E. Tarakanov, Combinatorial Problems and (0, 1)-Matrices [in Russian], Nauka, Moscow (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 14, No. 5, pp. 139–154, 2008.
Rights and permissions
About this article
Cite this article
Kupavskii, A.B., Raigorodskii, A.M. On the chromatic number of ℝ9 . J Math Sci 163, 720–731 (2009). https://doi.org/10.1007/s10958-009-9708-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-009-9708-4