The Borsuk problem for integral polytopes

Sbornik Mathematics

Volumn 193, Issue 9-10, 2002, Pages 1535-1556

  Raigorodskii A M ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036767848     PISSN: 10645616     EISSN: 14684802     Source Type: Journal    
DOI: 10.1070/sm2002v193n10abeh000688     Document Type: Article

References (51)

1. V. G. Boltyanski (Boltyanskiǐ) and I. Ts. Gohberg, Results and problems in combinatorial geometry, Nauka, Moscow 1965; English transl., Cambridge Univ. Press, Cambridge 1985.
2. V.G. Boltyanski (Boltyanskiǐ). H. Martini, and P. S. Soltan, "Excursions into combinatorial geometry. Universitext", Springer-Verlag, Berlin 1997.
3. A. M. Raǐgorodskiǐ, "The Borsuk problem and the chromatic numbers of some metric spaces", Uapekhi Mat. Nauk 56:1 (2001), 107-146; English transl. in Russian Math. Surveys 56 (2001).
4. K. Borsuk, "Drei Sätze über die n-dimensionale euklidische Sphäre", Fund. Math. 20 (1933), 177-190.
5. H. G. Eggleston, "Covering a three-dimensional set with sets of smaller diameter". J. London Math. Soc. (2) 30 (1955), 11-24.
6. J. Perkal, "Sur la subdivision des ensembles en parties de diamètre inférieur", Colloq. Math. 1 (1947), 45.
7. B. Grünbaum, "A simple proof of Borsuk's conjecture in three dimensions", Proc. Cambridge Philos. Soc. 53 (1957), 776-778.
8. A. Heppes, "Térbeli ponthalmazok felosztása kisebb átméröjü részhalmazok összegére", Magyar Tud. Akad. Math. Fix. Tud. Oszt. Közl 7 (1957), 413-416.
9. P. Katzarowa-Karanowa, "Über ein euklidisch-geometrisches Problem von B. Grünbaum", Arch. Math. (Basel) 18 (1967), 663-672.
10. V. V. Makeev, "On affine images of a rhombo-dodecahedron that are circumscribed about the three-dimensional convex body", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. 246 (1997), 191-195; English transl, in J. Math. Sci. (New York) 100 (2000).
11. V. V. Makeev, "On affine images of a rhombo-dodecahedron that are circumscribed about the three-dimensional convex body", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. 246 (1997), 191-195; English transl, in J. Math. Sci. (New York) 100 (2000).
12. V. V. Makeev, "Affinely inscribed and affinely circumscribed polygons and polyhedra", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. 231 (1996), 286-298; English transl. in J. Math. Sci. (New York) 91 (1998).
13. V. V. Makeev, "Affinely inscribed and affinely circumscribed polygons and polyhedra", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. 231 (1996), 286-298; English transl. in J. Math. Sci. (New York) 91 (1998).
14. H. Hadwiger, "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Comment. Math. Helv. 18 (1945/46). 73-75; "Mitteilung betreffend meine Note: "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Comment. Math. Helv. 10 (1946/47), 72-73.
15. H. Hadwiger, "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Comment. Math. Helv. 18 (1945/46). 73-75; "Mitteilung betreffend meine Note: "Überdeckung einer Menge durch Mengen kleineren Durchmessers", Comment. Math. Helv. 10 (1946/47), 72-73.
16. C. A. Rogers, "Symmetrical sets of constant width and their partitions", Mathemalika 18 (1971), 105-111.
17. A. S. Risling, "The Borsuk problem in 3-dimensional spaces of constant curvature", Ukrain. Geom. Sb. 11 (1971), 78-83. (Russian)
18. C. A. Rogers, "Covering a sphere with spheres", Mathematika 10 (1963), 157-164.
19. O. Schramm, "Illuminating sets of constant width", Mathematika 35 (1988), 180-189.
20. d by balls of the same diameter", Geometric aspects of functional analysis. Lecture Notes in Math., vol. 1469, Springer-Verlag, Berlin 1991, pp. 138-144.
21. J. Kahn and G. Kalai, "A counterexample to Borsuk's conjecture", Bull. Amer. Math. Soc. (N.S.) 20 (1993), 60-62.
22. A. Nilli, On Borsuk's problem, Contemporary Math. 178 (1994), 209-210.
23. A. M. Raǐgorodskiǐ, "On the dimension in Borsuk's problem", Uspekhi Mat. Nauk 52:6 (1997), 181-182; English transl, in Russian Math. Surveys 52 (1997).
24. A. M. Raǐgorodskiǒ, "On a bound in Borsuk's problem", Uspekhi Mat. Nauk 54:2 (1999), 185-186; English transl. In Russian Math. Surveys 54 (1999).
25. P. Erdös, "My Scottish book 'problems", The Scottish book. Mathematics from the Scottish Café, Birkhäuser, Boston 1981, pp. 35-43.
26. G. M. Ziegler, "Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions", Preprint, TU Berlin, June 2000.
27. C. Payan, "On the chromatic number of cube-like graphs", Discrete Math. 103 (1992), 271-277.
28. J. Petersen, "Färbung von Borsuk-Graphen in niedriger Dimension", Diplomarbeit, TU Berlin, 1998.
29. F. Schiller, "Zur Berechnung und Abschätzung von Färbungszahlen und der θ-Funklion von Graphen", Diplomarbeit, TU Berlin, 1999.
30. A. M. Ragǐorodskiǐ, "Borsuk's problem for (0, l)-polyhedra and cross-polytopes", Dokl. Ross. Akad. Nauk 371 (2000), 600-603; English transl, in Dokl. Math. 61 (2000).
31. A. M. Ragǐorodskiǐ, "Borsuk's problem for (0, l)-polyhedra and cross-polytopes", Dokl. Ross. Akad. Nauk 371 (2000), 600-603; English transl, in Dokl. Math. 61 (2000).
32. A. M. Raǐgorodskiǐ, "Borsuk's problem for (0, l)-polytopes and cross-polytopes", Dokl. Ross. Akad. Nauk 384 (2002), 593-597; English transl. in Dokl. Math. 65 (2002).
33. A. M. Raǐgorodskiǐ, "Borsuk's problem for (0, l)-polytopes and cross-polytopes", Dokl. Ross. Akad. Nauk 384 (2002), 593-597; English transl. in Dokl. Math. 65 (2002).
34. P. Erdös, "On sets of distances of n points", Amer. Math. Monthly 53 (1946), 248-250.
35. A. Heppes and P. Révész, "Zum Borsukschen Zerteilungsproblem", Acta Math. Acad. Sci. Hungar. 7 (1956), 159-162.
36. A. M. Raǐgorodskiǐ, "Systems of common representatives", Fund. Prikl. Math. 5 (1999), 851-860. (Russian)
37. N. N. Kuzyurin, "Asymptotic analysis of the set covering problem". Problemy Kibernet. 37 (1980), 19-56. (Russian)
38. P. Turán, "Egy grafelmeletiszelsoertek feladatrol", Mat. Fix. Lapok. 48 (1941), 436-452.
39. N. Linial. R. Meshulam, and M. Tarsi, "Matroidal bijections between graphs", J. Combin. Theory Ser. B 45:1 (1988), 31-44.
40. G. Katona, T. Nemetz, and M. Simonovitz, "On a graph-problem of Turán", Mat. Lapok 15 (1964), 228-238. (Hungarian)
41. Z. Füredi, "Turán's type problems", Surveys in combinatorics, London Math. Soc. Lecture Note Series, vol. 166, 1991, pp. 253-300.
42. A. M. Raǐgorodskiǐ, "The defects of admissible balls and octahedra in a lattice and systems of generic representatives", Mat. Sb. 189:6 (1998), 117-141: English transl, in Sb. Math. 189 (1998).
43. P. Erdös, Ch. Ko, and R. Rado, "Intersection theorems for systems of finite sets", Quart. J. Math. 12 (1961), 313-320.
44. R. Ahlswede and L. H. Khachatrian, "The complete non-trivial-intersection theorem for systems of finite sets", J. Combin. Theory Ser. A 76 (1996), 121-138.
45. R. Ahlswede and L. H. Khachatrian, "The complete intersection theorem for systems of finite sets", European J. Combin. 18 (1997), 125-136.
46. P. Frankl, "The Erdös-Ko-Rado theorem is true for n = ckt", Combinatorics. Colloq. Math. Soc. Jáos Bolyai 18 (1978), 365-375.
47. R. M. Wilson, "The exact bound in the Erdös-Ko-Rado theorem", Combinatorica 4 (1984), 247-257.
48. M. Deza and P. Frankl, "Erdös-Ko-Rado theorem - 22 years later", SIAM J. Algebraic Discrete Methods 4 (1983), 419-431.
49. A. M. Raǐgorodskiǐ, "The Borsuk partition problem", London Math. Soc. Lecture Note Series (to appear).
50. A. M. Raǐgorodskiǐ and Yu. A. Kalnishkan, "On the Borsuk problem", Modern problems in number theory and its applications (Tula, September 10-15, 2001), Tula pp. 98-99. (Russian)
51. M. Lassak, "An estimate concerning Borsuk's partition problem", Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), 449-451.