## On rainbow isosceles $$n$$-simplexes.(English)Zbl 1498.52023

The paper is dedicated to geometric problems connected with various colorings of all points of an Euclidean space $$\mathbb{R}^n$$ for certain types of $$(n+1)$$-colorings of space. It is proved that there are continuum many “rainbow” isosceles $$n$$-simplexes in $$\mathbb{R}^n$$.

### MSC:

 52B45 Dissections and valuations (Hilbert’s third problem, etc.) 51M04 Elementary problems in Euclidean geometries 51M20 Polyhedra and polytopes; regular figures, division of spaces
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### References:

 [1] M. Aigner and G. M. Ziegler, Proofs from The Book, 3rd ed., Springer, Berlin, 2004. · Zbl 1098.00001 [2] D. Coulson, A 15-colouring of 3-space omitting distance one, Discrete Math. 256 (2002), no. 1-2, 83-90. · Zbl 1007.05052 [3] L. Danzer, B. Grünbaum and V. Klee, Helly’s theorem and its relatives, Proceedings of Symposia in Pure Mathematics. Vol. 7 American Mathematical Society, Providence (1963), 101-180. · Zbl 0132.17401 [4] A. D. N. J. de Grey, The chromatic number of the plane is at least 5, Geombinatorics 28 (2018), no. 1, 18-31. · Zbl 1404.05063 [5] R. Engelking, General Topology, PWN, Warszawa, 1985. [6] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957. · Zbl 0078.35703 [7] H. Hadwiger and H. Debrunner, Combinatorial Geometry in the Plane, Holt, Rinehart and Winston, New York, 1964. [8] A. B. Harazišvili, Orthogonal simplexes in four-dimensional space (in Russian), Bull. Acad. Sci. Georgian SSR 88 (1977), no. 1, 33-36. · Zbl 0382.52004 [9] A. Kharazishvili, Three-colorings of the Euclidean plane and associated triangles of a prescribed type, Georgian Math. J. 22 (2015), no. 3, 393-396. · Zbl 1320.05043 [10] K. Kuratowski, Topology. Vol. I, Academic Press, New York, 1966. [11] D. G. Mead, Dissection of the hypercube into simplexes, Proc. Amer. Math. Soc. 76 (1979), no. 2, 302-304. · Zbl 0423.51012 [12] P. Monsky, On dividing a square into triangles, Amer. Math. Monthly 77 (1970), 161-164. · Zbl 0187.19701 [13] O. Nechushtan, On the space chromatic number, Discrete Math. 256 (2002), no. 1-2, 499-507. · Zbl 1009.05058 [14] V. Pambuccian, Existence of special rainbow triangles in weak geometries, Georgian Math. J. 26 (2019), no. 4, 489-498. · Zbl 1431.51007 [15] J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921), no. 1-2, 113-115. · JFM 48.0834.04 [16] A. M. Raigorodskii, Chromatic Numbers (in Russian), Moscow Centre for Continuing Mathematics Education, Moscow, 2003. · Zbl 1125.05041 [17] S. Shelah and A. Soifer, Axiom of choice and chromatic number of the plane, J. Combin. Theory Ser. A 103 (2003), no. 2, 387-391. · Zbl 1030.03035 [18] K. Tschirpke, On the dissection of simplices into orthoschemes, Geom. Dedicata 46 (1993), no. 3, 313-329. · Zbl 0780.52016 [19] K. Tschirpke, The dissection of five-dimensional simplices into orthoschemes, Beitr. Algebra Geom. 35 (1994), 1-11. · Zbl 0806.52012
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