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chromatic number of a metric space∗ Mathprof† 2013-03-21 16:56:04 The chromatic number of a metric space is the minimum number of colors required to color the space in such a way that no two points at distance 1 are assigned the same color. Alternatively, the chromatic number of a metric space is the chromatic number of a graph whose vertices are points of the space, and two points are connected by an edge if they are at distance 1 from each other. For example, the chromatic number of R is 2 because it is impossible to color R into one color, but it is possible to color R into 2 colors by coloring each of the intervals [k, k + 1), where k is an integer, red or blue according to whether k is odd or even. Unlike R, the chromatic number of Rn is not known for any n ≥ 2. For example, the chromatic number of the plane is known to be between 4 and 7 as the following pictures show. 7 • • • 7 1 • 3 5 3 5 7 2 7 3 1 6 1 6 4 2 4 2 7 4 7 5 3 5 3 1 5 1 6 4 6 • 4 2 2 4 6 4 1 • • The Moser spindle. All edges are of unit length. The chromatic number is 4. Periodic coloring of the plane with 7 colors. The diameter of each hexagonal region is slightly less than 1. ∗ hChromaticNumberOfAMetricSpacei created: h2013-03-21i by: hMathprofi version: h35510i Privacy setting: h1i hDefinitioni h05C15i h52C10i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 Using compactness argument one can show that if every finite set of points in the plane can be colored with 4 colors, then the whole plane can be colored with 4. However, if such a coloring exists, then at least one of the color sets is nonmeasurable with respect to the Lebesgue measure [?]. If the coloring of the plane is to consist of regions bounded by Jordan curves, then at least 6 colors are needed [?]. Moreover, under the additional assumption that no two regions, which are distance less than 1 apart, receive the same color, then at least 7 colors are needed [?]. The following table summarizes known lower and upper bounds on the chromatic number of some metric spaces. R2 R3 R4 R5 R6 R7 Rn Upper bound 7 15[?, ?] (3 + o(1))n [?] Lower bound 4 6[?] 7[?] 9[?] 10[?] 15[?] (1.239 + o(1))n [?, ?] Q2 Q3 Upper bound 2[?] 2[?] Lower bound 2[?] 2[?] For more information on the see [?]. Q4 Q5 Q6 Q7 Qn 4[?] 4[?] 7[?] 10[?] 13[?] (1.173 + o(1))n [?] lower bounds for the chromatic numbers of Rn References [1] M. Benda and M. Perles. Colorings of metric spaces. Geombinatorics, 9(3):113–126, 2000. Zbl 0951.05037. [2] David Coulson. A 15-colouring of 3-space omitting distance one. Discrete Math., 256:83–90, 2002. Zbl 1007.05052. [3] K. J. Falconer. The realization of distances in measurable subsets covering Rn . J. Combin. Theory Ser. A, 31:184–189, 1981. Zbl 0469.05021. [4] Peter D. Johnson. Coloring abelian groups. Discrete Math., 40:219–223, 1982. Zbl 0485.05009. [5] D. G. Larman and C. A. Rogers. The realization of distances within sets in Euclidean space. Mathematika, 19:1–24, 1972. Zbl 0246.05020. [6] Matthias Mann. A new bound for the chromatic number of the rational five-space. Geombinatorics, 11(2):49–53, 2001. Zbl 0995.05051. [7] Matthias Mann. Hunting unit-distance graphs in rational n-spaces. Geombinatorics, 13(2):86–97, 2003. [8] Radoš Radoičić and Géza Tóth. Note on the chromatic number of the space. In Discrete and computational geometry, volume 25 of Algorithms Combin., pages 695–698. Springer, Berlin, 2003. Zbl 1071.05527. [9] Andrei M. Raigorodskii. On the chromatic number of a space. Russian Math. Surveys, 55(2):351–352, 2000. Zbl 0966.05029. 2 [10] Andrei M. Raigorodskii. Borsuk’s problem and the chromatic numbers of some metric spaces. Russian Math. Surveys, 56(1):103–139, 2001. Zbl 1008.54018. [11] Andrei M. Raigorodskii. Available at http://www.mccme.ru/mmmflectures/books/index.php?task=video, 2002. [12] D. E. Raiskii. The realization of all distances in a decomposition of the space Rn into n + 1 parts. Math. Notes, 7:194–196, 1970. Zbl 0202.21702. [13] László A. Székely. Erdős on unit distances and Szemerédi-Trotter theorems. Preprint is at http://www.math.sc.edu/ szekely/, 2002. [14] Carsten Thomassen. On the Nelson unit distance coloring problem. Amer. Math. Monthly, 106(9):850–853, 1999. Zbl 0986.05041. Available online at JSTOR. [15] D. R. Woodall. Distances realized by sets covering the plane. J. Combin. Theory Ser. A, 14:187–200, 1973. Zbl 0251.50003. [16] K. Cantwell. Finite Euclidean Ramsey theory. J. Combin. Theory Ser. A, 73:273–285, 1996. Zbl 0842.05064. [17] O. Neuchushtan. On the space chromatic number. Discrete Mathematics. 256:499–507, 2002. Zbl 1009.05058. 3