Pach’s Selection Theorem Does Not Admit a Topological Extension

  • Imre Bárány
  • Roy Meshulam
  • Eran Nevo
  • Martin Tancer
Article
  • 7 Downloads

Abstract

Let \(U_1,\ldots ,U_{d+1}\) be n-element sets in \(\mathbb {R}^d\). Pach’s selection theorem says that there exist subsets \(Z_1\subset U_1,\ldots ,Z_{d+1}\subset U_{d+1}\) and a point \(u \in \mathbb {R}^d\) such that each \(|Z_i|\ge c_1(d)n\) and \(u \in {{\mathrm{conv}}}\{ z_1,\ldots ,z_{d+1}\}\) for every choice of \(z_1 \in Z_1,\ldots ,z_{d+1} \in Z_{d+1}\). Here we show that this theorem does not admit a topological extension with linear size sets \(Z_i\). However, there is a topological extension where each \(|Z_i|\) is of order \((\log n)^{1/d}\).

Keywords

Pach’s Selection Theorem Gromov’s Overlap Theorem 

Mathematics Subject Classification

52A35 52C99 

Notes

Acknowledgements

This research was supported by ERC Advanced Research Grant no 267165 (DISCONV). Imre Bárány is partially supported by Hungarian National Research Grant K 111827. Roy Meshulam is partially supported by ISF grant 326/16 and GIF grant 1261/14, Eran Nevo by ISF grant 1695/15 and Martin Tancer by GAČR grant 16-01602Y.

References

  1. 1.
    Alon, N.: Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6(3), 207–219 (1986)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2–3), 141–152 (1982)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bárány, I., Füredi, Z., Lovász, L.: On the number of halving planes. Combinatorica 10, 175–183 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boros, E., Füredi, Z.: The number of triangles covering the center of an \(n\)-set. Geom. Dedicata. 17(1), 69–77 (1984)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bukh, B., Hubard, A.: On a topological version of Pach’s overlap theorem (2017). https://arxiv.org/abs/1708.04350
  6. 6.
    Corrádi, K.: Problem at the Schweitzer competition. Mat. Lapok 20, 159–162 (1969)Google Scholar
  7. 7.
    Dotterrer, D., Kahle, M.: Coboundary expanders. J. Topol. Anal. 4(4), 499–514 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdős, P.: On extremal problems of graphs and generalized graphs. Israel J. Math. 2, 183–190 (1964)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gromov, M.: Singularities, expanders and topology of maps. Part 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20(2), 416–526 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lovász, L.: Combinatorial Problems and Exercises, 2nd edn. North-Holland, Amsterdam (1993)MATHGoogle Scholar
  11. 11.
    Lubotzky, A., Meshulam, R., Mozes, S.: Expansion of building-like complexes. Groups Geom. Dyn. 10(1), 155–175 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Pach, J.: A Tverberg-type result on multicolored simplices. Comput. Geom. 10(1), 71–76 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Stanley, R.P.: Enumerative Combinatorics. Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2012)Google Scholar
  14. 14.
    Tancer, M.: Non-representability of finite projective planes by convex sets. Proc. Amer. Math. Soc. 138(9), 3285–3291 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Rényi InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Department of MathematicsUniversity College LondonLondonUK
  3. 3.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael
  4. 4.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  5. 5.Department of Applied MathematicsCharles University in PraguePraha 1Czech Republic

Personalised recommendations