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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 813–831 | Cite as

Random Steiner systems and bounded degree coboundary expanders of every dimension

  • Alexander Lubotzky
  • Zur LuriaEmail author
  • Ron Rosenthal
Article
  • 166 Downloads

Abstract

We introduce a new model of random d-dimensional simplicial complexes, for \(d\ge 2\), whose \((d-1)\)-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash’s recent result on designs (The existence of designs; arXiv:1401.3665, 2014), and the proof of the expansion uses techniques developed by Evra and Kaufman in (Bounded degree cosystolic expanders of every dimension; arXiv:1510.00839, 2015). This gives a full solution to a question raised in Dotterrer and Kahle (J Topol Anal 4(4): 499–514, 2012), which was solved in the two-dimensional case by Lubotzky and Meshulam (Adv Math 272: 743–760, 2015).

Keywords

Simplicial complexes Designs Steiner systems Coboundary expansion 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their suggestions which helped to improve this manuscript.

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

Authors and Affiliations

  1. 1.Einstein Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Institute of Theoretical StudiesETH ZürichZürichSwitzerland
  3. 3.Israel Institute for Advanced StudiesHebrew University of Jerusalem, Givat RamJerusalemIsrael
  4. 4.Departement MathematikETH ZürichZürichSwitzerland
  5. 5.Technion - Israel Institute of TechnologyHaifaIsrael

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