ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 243–250. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © R. I. Prosanov, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 248–257.
Upper Bounds
for the Chromatic Numbers of Euclidean Spaces
with Forbidden Ramsey Sets
R. I. Prosanov
*
Lomonosov Moscow State University, Moscow, Russia
Received October 1, 2016; in final form, February 7, 2017
Abstract—The chromatic number of a Euclidean space R
n
with a forbidden finite set C of points is
the least number of colors required to color the points of this space so that no monochromatic set is
congruent to C. New upper bounds for this quantity are found.
DOI: 10.1134/S000143461801025X
Keywords: Euclidean Ramsey theory, chromatic number of space.
1. INTRODUCTION
We de fine the chromatic number χ(R
n
) of Euclidean space R
n
as the least number of colors required
to color all points of this space so that any two points a distance 1 apart are of different colors. The
problem of finding the chromatic number was first stated by Nelson in 1950 for R
2
(see [1]–[4] for details).
The exact value of the chromatic number is unknown even for the Euclidean plane. At present, the
best estimates are
4 ≤ χ(R
2
) ≤ 7
(see [1]). For the case of multidimensional Euclidean space, it has been proved that
(1.239 + o(1))
n
≤ χ(R
n
) ≤ (3 + o(1))
n
;
the lower bound was found by Raigorodskii in [5], and the upper bound, by Larman and Rogers in [6].
Consider a more general problem. Suppose given a set of points C ⊂ R
d
. We want to determine the
least number χ(C, n) of colors in a coloring of the points of R
n
(n ≥ d) such that no monochromatic sets
are congruent to C. Clearly, the problem of finding the chromatic number of space equivalent to that
stated above with d =1and C = {0, 1}.Moreover,foranyC containing more than one point, we have
χ(C, n) ≤ χ(R
n
) ≤ (3 + o(1))
n
.
We say that a set C is Ramsey if χ(C, n) tends to infinity with increasing n.
It is easy to show that only finite sets can be Ramsey. Another known necessary condition for a set
to be Ramsey is that any Ramsey set lies on a sphere (see [7]). The central conjecture of Euclidean
Ramsey theory is that these two conditions are sufficient (see [7]). Since the lower bound for the
chromatic number of n-space increases exponentially with dimension, it follows that the unit interval is
a Ramsey set. It has also been proved that all simplices [8], [9], d-hypercubes [7], regular polygons [10],
and cross-polytopes [11] are Ramsey sets; moreover, for simplices and hypercubes, χ(C, n) increases
exponentially with n (see [9], [12]–[18]).
In this paper, we improve the upper bound for the quantity χ(C, n) as depending on certain
parameters of the set C. As corollaries, we obtain upper bounds for this quantity for all of the sets
mentioned above.
*
E-mail: rprosanov@mail.ru
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