Skip to main content
SpringerLink
Log in

Strong ramsey properties of simplices

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper we will show that every simplexX with circumradiusϱ satisfies the following geometric partition property, which proves a conjecture from [FR90].

For every positive realδ there exists a positive realσ such that everygc-colouring of then-dimensional sphere of radiusϱ+δ withχ≤(1+σ)n results in a monochromatic copy ofX.

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

  1. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus,Euclidean Ramsey theorems. I. Journal of Combinatorial Theory. Series A14 (1973), 341–363.

    Article  Google Scholar 

  2. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus,Euclidean Ramsey theorems. II, inInfinite and Finite Sets (Colloquium, Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, Colloquia Mathematica Societatis János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 529–557.

    Google Scholar 

  3. P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer and E. G. Straus,Euclidean Ramsey theorems, III, inInfinite and Finite Sets (Colloquium, Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I, Colloquia Mathematica Societatis János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 559–583.

    Google Scholar 

  4. P. Frankl and V. Rödl,Forbidden intersections, Transactions of the American Mathematical Society300 (1987), 259–286.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Frankl and V. Rődl,A partition property of simplices in Euclidean space, Journal of the American Mathematical Society3 (1990), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Frankl,Intersection theorems for finite sets and geometric applications, inProceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, California, 1986), American Mathematical Society, Providence, RI, 1987, pp. 1419–1430.

    Google Scholar 

  7. P. Frankl and R. M. Wilson,Intersection theorems with geometric consequences, Combinatorica1 (1981), 357–368.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. L. Graham,Euclidean Ramsey theorems on the n-sphere, Journal of Graph Theory7 (1983), 105–114.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. L. Graham,Old and new Euclidean Ramsey theorems, inDiscrete Geometry and Convexity (New York, 1982), New York Academy of Science, New York, 1985, pp. 20–30.

    Google Scholar 

  10. R. L. Graham,Recent trends in Euclidean Ramsey theory, Discrete Mathematics136 (1994), 119–127.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Hadwiger,Ungelöste probleme no. 40, Elements in Mathematics16 (1961), 103–104.

    MathSciNet  Google Scholar 

  12. I. Kříž,Permutation groups in Euclidean Ramsey theory, Proceedings of the American Mathematical Society112 (1991), 899–907.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. G. Larman and C. A. Rogers,The realization of distances within sets in Euclidean space, Mathematika19 (1972), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Moser and W. Moser,Solution to problem 10, Canadian Mathematical Bulletin4 (1961), 187–189.

    Google Scholar 

  15. J. Matoušek and V. Rödl,On Ramsey sets in spheres, Journal of Combinatorical Theory. Series A70 (1995), 30–44.

    Article  MATH  Google Scholar 

  16. A. M. Raigorodskii,On the chromatic number of a space, Uspekhi Matematicheskikh Nauk55 (2000), no. 2(332), 147–148; English translation: Russian Mathematical Surveys55 (2000), no. 2, 351–352.

    MathSciNet  Google Scholar 

  17. A. M. Raigorodskii,The Borsuk problem and the chromatic numbers of some metric spaces, Uspekhi Matematicheskikh Nauk56 (2001), no. 1(337), 107–146; English translation: Russian Mathematical Surveys56 (2001), no. 1, 103–139.

    MathSciNet  Google Scholar 

  18. V. Rödl,On a problem in combinatorial geometry, Discrete Mathematics45 (1983), 129–131.

    Article  MATH  MathSciNet  Google Scholar 

  19. I. J. Schoenberg,Metric spaces and positive definite functions, Transactions of the American Mathematical Society44 (1938), 522–536.

    Article  MATH  MathSciNet  Google Scholar 

  20. D. R. Woodall,Distances realized by sets covering the plane, Journal of Combinatorial Theory. Series A14 (1973), 187–200.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Shibuya-ku, Higashi 1-14-4, 150-0011, Tokyo, Japan

    Peter Frankl

  2. Department of Mathematics and Computer Science, Emory University, 30322, Atlanta, GA, USA

    Vojtěch Rödl

Authors
  1. Peter Frankl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vojtěch Rödl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Frankl.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frankl, P., Rödl, V. Strong ramsey properties of simplices. Isr. J. Math. 139, 215–236 (2004). https://doi.org/10.1007/BF02787550

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02787550

Keywords

Search

Navigation