Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

A counterexample to Borsuk's conjecture

Author(s): Jeff Kahn; Gil Kalai
Journal: Bull. Amer. Math. Soc. 29 (1993), 60-62.
MSC (2000): Primary 52A20
MathSciNet review: 1193538
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.

References:

[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.

Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 52A20

Retrieve articles in all Journals with MSC (2000): 52A20

Additional Information:

DOI: 10.1090/S0273-0979-1993-00398-7
PII: S 0273-0979(1993)00398-7
Copyright of article: Copyright 1993, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia