Pach’s Selection Theorem Does Not Admit a Topological Extension

  • Imre Bárány
  • Roy Meshulam
  • Eran Nevo
  • Martin Tancer


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 \((\log n)^{1/d}\).


Pach’s Selection Theorem Gromov’s Overlap Theorem 

Mathematics Subject Classification

52A35 52C99 



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.


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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

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