The Borsuk partition problem: The seventieth anniversary

Mathematical Intelligencer

Volumn 26, Issue 3, 2004, Pages 4-12

  Raǐgorodskiǐ, Andreǐ M ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 4243180293     PISSN: 03436993     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02986745     Document Type: Review

References (39)

1. K. Borsuk, Drei Sätze über die n-dimensionale Euklidische Sphäre, Fundaments Math., 20 (1933), 177-190.
2. F. Reuleaux, Lehrbuch der Kinematik I. Vieweg, Braunschweig 1875 (Engl. transl.: 1876; Reprint: Dover, New York 1963).
3. H. G. Eggleston, Convexity, Cambridge Univ. Press, Cambridge, 1958.
4. K. Borsuk, Über die Zerlegung einer Euklidischen n-dimensionalen Vollkugel in n Mengen, Verh. Internat. Math. Kongr., Zürich, 2 (1932), 192.
5. L. A. Lyusternik and L. G. Shnirel'man, Topological methods for variational problems, ONTI, Moscow, 1930.
6. H. Lenz, Zur Zerlegung von Punktmengen in solche kleineren Durchmessers, Archiv Math., 6 (1955), N 5, 413-416.
7. H. Hadwiger, Überdeckung einer Menge durch Mengen kleineren Durchmessers, Comm. Math. Helv., 18 (1945/46), 73-75; Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers, Comm. Math. Helv., 19 (1946/47), 72-73.
8. H. Hadwiger, Überdeckung einer Menge durch Mengen kleineren Durchmessers, Comm. Math. Helv., 18 (1945/46), 73-75; Mitteilung betreffend meine Note: Überdeckung einer Menge durch Mengen kleineren Durchmessers, Comm. Math. Helv., 19 (1946/47), 72-73.
9. V. G. Boltyanskiǐ, I. Ts. Gokhberg, Results and problems in combinatorial geometry, Nauka, Moscow, 1965; English transl., Cambridge Univ. Press, Cambridge, 1985.
10. M. Lassak, An estimate concerning Borsuk's partition problem, Bull. Acad. Polon. Sci. Ser. Math., 30 (1982), 449-451.
11. C. A. Rogers, Covering a sphere with spheres, Mathematika, 10 (1963), 157-164.
12. O. Schramm, Illuminating sets of constant width, Mathematika, 35 (1988), 180-189.
13. d by balls of the same diameter, Geometric Aspects of Functional Analysis (J. Lindenstrauss and V. Milman, eds.), Lecture Notes in Math., vol. 1469, Springer-Verlag, Berlin, 1991, 138-144.
14. P. Erdos, My Scottish book "problems", The Scottish Book, Mathematics from the Scottish Café (R.D. Mauldin ed.), Birkhäuser, 1981, 35-43.
15. C. A. Rogers, Symmetrical sets of constant width and their partitions, Mathematika, 18 (1971), 105-111.
16. D. Larman, Open problem 6, Convexity and Graph theory (M. Rozenfeld and J. Zaks eds.), Ann. Discrete Math., vol. 20, North-Holland, Amsterdam and New York, 1984, p. 336.
17. J. Kahn and G. Kalai, A counterexample to Borsuk's conjecture, Bulletin (new series) of the AMS, 29 (1993), N 1, 60-62.
18. A. Hinrichs and Ch. Richter, New sets with large Borsuk numbers, 2002, http://www.minet.uni-jena.de/hinrichs/paper/18/borsuk.pdf.
19. A. M. Raǐgorodskiǐ, On a bound in Borsuk's problem, Uspekhi Mat. Nauk, 54(1999), N 2, 185-186; English transl., Russian Math. Surveys, 54 (1999), 453-454.
20. A. M. Raǐgorodskiǐ, On a bound in Borsuk's problem, Uspekhi Mat. Nauk, 54(1999), N 2, 185-186; English transl., Russian Math. Surveys, 54 (1999), 453-454.
21. G. M. Ziegler, Coloring Hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions, Lect, Notes Comput. Sci., 2122 (2001), 159-171.
22. A. M. Raǐgorodskiǐ, The Borsuk problem and the chromatic numbers of some metric spaces, Uspekhi Mat. Nauk, 56 (2001 ), N 1, 107-146; English transl., Russian Math. Surveys, 56 (2001), 103-139.
23. A. M. Raǐgorodskiǐ, The Borsuk problem and the chromatic numbers of some metric spaces, Uspekhi Mat. Nauk, 56 (2001 ), N 1, 107-146; English transl., Russian Math. Surveys, 56 (2001), 103-139.
24. V. G. Boltyanski, H. Martini, and P. S. Soltan, Excursions into combinatorial geometry, Universitext, Springer-Verlag, Berlin Heidelberg, 1997.
25. H. Hadwiger, H. Debrunner, V. Klee, Combinatorial geometry in the plane, Holt, Rinehart and Winston, New York, 1964.
26. L. Danzer, B. Grünbaum, V. Klee, Helly's theorem and its relatives, Convexity (V. Klee, ed.), Providence, R.I., 1963, 99-180.
27. H. G. Eggleston, Covering a three-dimensional set with sets of smaller diameter, J. London Math, Soc., 30 (1955), 11-24.
28. A. Heppes, Térbeli ponthalmazok felosztása kisebb átméröjü részhalmazok összegére, A magyar tudományos akadémia, 7 (1957), 413-416.
29. B. Grünbaum, A simple proof of Borsuk's conjecture in three dimensions, Proc. Cambridge Philos. Soc., 53 (1957), 776-778.
30. 3, Zapiski Nauchn. Semin, POMI, 246 (1997), 191-195; English transl., J. Math. Sci. (New York), 100 (2000), 2307-2309.
31. 3, Zapiski Nauchn. Semin, POMI, 246 (1997), 191-195; English transl., J. Math. Sci. (New York), 100 (2000), 2307-2309.
32. 3, Mat. Zametki, 74 (2003), N 1, 149-151; English transi., Math. Notes, 2003.
33. 3, Mat. Zametki, 74 (2003), N 1, 149-151; English transi., Math. Notes, 2003.
34. P. Erdos, 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. Hung., 7 (1956), 159-162.
36. V. V. Makeev, Universal coverings and projections of bodies of constant width, Ukrain, Geom. Sb., 32 (1989), 84-88 (in Russian).
37. B. Weissbach, Polyhedral covers, Coll, Math. Soc. J. Bolyai 48 (Intuitive geometry), North-Holland, Amsterdam 1987, 639-646.
38. M. Aigner and G. M. Ziegler, Proofs from THE BOOK, Springer-Verlag, Berlin, 1998.
39. P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica, 1 (1981), 357-368.