Skip to main content
SpringerLink
Log in

Borsuk’s partition problem and finite point sets

  • Original Paper
  • Published:
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Aims and scope Submit manuscript

Abstract

Borsuk’s partition problem in \(\mathbb {R}^n\) can be rephrased in terms of finite point sets, in fact in terms of finite subsets of \(\mathbb {Z}^n\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Berger, M.: Geometry I. Springer, Berlin (1987)

    Book  Google Scholar 

  • Bondarenko, A.V.: On Borsuk’s conjecture for two-distance sets. Discrete Comput. Geom. 51, 509–515 (2014)

    Article  MathSciNet  Google Scholar 

  • Borsuk, K.: Drei Sätze über die \(n\)-dimensionale euklidische Sphäre. Fund. Math. 20, 177–190 (1933)

    Article  Google Scholar 

  • Burkert, J.: Explicit colorings of \({\mathbb{Z}}^3\) and \({\mathbb{Z}}^4\) with four colors to forbid arbitrary distances. Geombinatorics 18, 149–152 (2009)

    MathSciNet  Google Scholar 

  • Cowen, R.H.: Two hypergraph theorems equivalent to BPI. Notre Dame J. Formal Logic 31, 232–240 (1990)

    Article  MathSciNet  Google Scholar 

  • De Bruijn, N.G., Erdös, P.: A colour problem for infinite graphs and a problem in the theory of relations. Nederl. Akad. Wetensch. Proc. Ser. A 54(Indagationes Math. 13), 369–373 (1951)

    MathSciNet  MATH  Google Scholar 

  • Erdős, P., Purdy, G.: Extremal problems in combinatorial geometry. In: Graham, R., et al. (eds.) Handbook of Combinatorics, Chapter 17, pp. 809–874. Elsevier, Amsterdam (1995)

    Google Scholar 

  • Grünbaum, B.: Borsuk’s problem and related questions. In: Convexity, Proc. Sympos. Pure Math., vol. VII, pp. 271–284. Amer. Math. Soc., Providence (1963)

  • Hadwiger, H., Debrunner, H.: Combinatorial Geometry in the Plane. Holt, Rinehart and Winston, London (1964), Dover edition (2015)

  • Jenrich, T., Brouwer, A.E.: A \(64\)-dimensional counterexample to Borsuk’s conjecture. Electron. J. Comb. 21, #P4.29 (2014)

    MathSciNet  MATH  Google Scholar 

  • Johnson, P.: \(4 = B_1({\mathbb{Q}}^3) = B_1({\mathbb{Q}}^4)\)!. Geombinatorics 17, 117–123 (2008)

    MathSciNet  Google Scholar 

  • Kahn, J., Kalai, G.: A counterexample to Borsuk’s conjecture. Bull. Am. Math. Soc. 29, 60–62 (1993)

    Article  MathSciNet  Google Scholar 

  • Kalai, G.: Some old and new problems in combinatorial geometry I: Around Borsuk’s problem. In: Czumaj, A., Georgakopoulos, A., Král, D., Lozin, V., Pikhurko, O. (eds.) Surveys in Combinatorics 2015. London Mathematical Society Lecture Note Series, pp. 147–174. Cambridge University Press, Cambridge (2015). https://doi.org/10.1017/CBO9781316106853.005

  • Komjáth, P., Totik, V.: Problems and Theorems in Classical Set Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Kuratowski, K.: Topology, Volumes 1 and 2. Academic Press, New York (1966, 1968)

    Chapter  Google Scholar 

  • Läuchli, H.: Coloring infinite graphs and the Boolean prime ideal theorem. Israel J. Math. 9, 422–429 (1971)

    Article  MathSciNet  Google Scholar 

  • Lenz, H.: Über die Bedeckung ebener Punktmengen durch solche kleineren Durchmessers. Arch. Math. 7, 34–40 (1956)

    Article  MathSciNet  Google Scholar 

  • Lovász, L.: Combinatorial Problems and Exercises. North Holland, Amsterdam (1979)

    MATH  Google Scholar 

  • Manturov, V.O.: On the chromatic numbers of integer and rational lattices. J. Math. Sci. (NY) 214, 687–698 (2016)

    Article  MathSciNet  Google Scholar 

  • Raigorodskii, A.: Three lectures on the Borsuk partition problem. In: Young, N., Choi, Y. (eds.) Surveys in Contemporary Mathematics. London Mathematical Society Lecture Note Series, pp. 202–247. Cambridge University Press, Cambridge (2007). https://doi.org/10.1017/CBO9780511666315.007

  • Raiskii, D.E.: Realization of all distances in a decomposition of the space \({\mathbb{R}}^n\) into \(n+1\) parts. Math. Notes 7, 194–196 (1970)

    Article  MathSciNet  Google Scholar 

  • Soifer, A.: The Mathematical Coloring Book. Springer, Berlin (2009)

    MATH  Google Scholar 

  • Wagner, K.: Graphentheorie. Bibliographisches Institut, Mannheim (1970)

    MATH  Google Scholar 

  • Woodall, D.R.: Distances realized by sets covering the plane. J. Comb. Theory Ser. A 14, 187–200 (1973)

    Article  MathSciNet  Google Scholar 

  • Ziegler, G.M.: Coloring hamming graphs, optimal binary codes, and the 0/1-Borsuk problem in low dimensions. In: Alt, H. (ed.) Computational Discrete Mathematics. Lecture Notes in Computer Science, vol. 2122, pp. 159–171. Springer, Berlin, Heidelberg (2001)

    Chapter  Google Scholar 

  • Zong, C.: A quantitative program for Hadwiger’s covering conjecture. Sci. China Math. 53, 2551–2560 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Günter Ziegler for comments and references.

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Emil-Fischer-Str. 30, 97074, Würzburg, Germany

    Theo Grundhöfer

Authors
  1. Theo Grundhöfer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Theo Grundhöfer.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grundhöfer, T. Borsuk’s partition problem and finite point sets. Beitr Algebra Geom 59, 709–716 (2018). https://doi.org/10.1007/s13366-018-0390-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13366-018-0390-x

Keywords

Mathematics Subject Classification

Search

Navigation