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A counterexample to Borsuk's conjecture


Authors: Jeff Kahn and Gil Kalai
Journal: Bull. Amer. Math. Soc. 29 (1993), 60-62
MSC (2000): Primary 52A20
DOI: https://doi.org/10.1090/S0273-0979-1993-00398-7
MathSciNet review: 1193538
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Abstract: Let $ f(d)$ be the smallest number so that every set in $ {R^d}$ of diameter 1 can be partitioned into $ f(d)$ sets of diameter smaller than 1. Borsuk's conjecture was that $ f(d) = d + 1$. We prove that $ f(d) \geq (1.2)\sqrt d $ for large d.


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  • [1] V. Boltjansky and I. Gohberg, Results and problems in combinatorial geometry, Cambridge Univ. Press, Cambridge, 1985. MR 821465 (87c:52002)
  • [2] K. Borsuk, Drei Sätze über die n-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
  • [3] J. Bourgain and J. Lindenstrauss, On covering a set in $ {R^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, pp. 138-144. MR 1122618 (92g:52018)
  • [4] H. Croft, K. Falconer, and R. Guy, Unsolved problems in geometry, Springer-Verlag, New York, 1991, pp. 123-125. MR 1107516 (92c:52001)
  • [5] L. Danzer, On the k-th diameter in $ {E^d}$ and a problem of Grünbaum, Proc. Colloq. on Convexity 1965 (W. Fenchel, ed.), Kø benhavns Univ. Math. Inst., 1967.
  • [6] P. Erdős, My Scottish book "problems", The Scottish Book, Mathematics from the Scottish Café (R. D. Mauldin, ed.), Birkhäuser, 1981, pp. 35-43. MR 666400 (84m:00015)
  • [7] P. Frankl and V. Rödl, Forbidden intersections, Trans. Amer. Math. Soc. 300 (1987), 259-286. MR 871675 (88m:05003)
  • [8] P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357-368. MR 647986 (84g:05085)
  • [9] B. Grünbaum, Borsuk's problem and related questions, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc, Providence, RI, 1963.
  • [10] J. Kahn and G. Kalai, A problem of Füredi and Seymour on covering intersecting families by pairs (to appear). MR 1297176 (96b:05126)
  • [11] J. Kahn and P. Seymour, A fractional version of the Erdős-Faber-Lovász conjecture, Combinatorica 12 (1992), 155-160. MR 1179253 (93g:05108)
  • [12] D. Larman and C. Rogers, The realization of distances within sets in Euclidean space, Mathematika 19 (1972), 1-24. MR 0319055 (47:7601)
  • [13] 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.
  • [14] M. Lassak, An estimate concerning Borsuk's partition problem, Bull. Acad. Polon. Sci. Ser. Math. 30 (1982),449-451. MR 703571 (84j:52014)
  • [15] C. A. Rogers, Some problems in the geometry of convex bodies, The Geometric Vein--The Coxeter Festschrift (C. Davis, B. Grünbaum, and F. A. Sherk, eds.), Springer-Verlag, New York, 1981, pp. 279-284. MR 661785 (84b:52001)
  • [16] O. Schramm, Illuminating sets of constant width, Mathematika 35 (1988), 180-199. MR 986627 (89m:52013)
  • [17] M. Simonovits and V. Sós, Graph intersection theorems. II, Combinatorics (A. Hajnal and V. Sós, eds.), North-Holland, Amsterdam, 1978, pp. 1017-1030.

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Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1993-00398-7
Article copyright: © Copyright 1993 American Mathematical Society

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