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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
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•
•
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
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4
2
2
4
6
4
1
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•
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.
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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
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