Cutting lemma and Zarankiewicz’s problem in distal structures

• Artem Chernikov ^[1] ; David Galvin ^[2] ; Sergei Starchenko ^[2]
  1. [1] University of California Los Angeles

     University of California Los Angeles

     Estados Unidos

     
  2. [2] University of Notre Dame

     University of Notre Dame

     Township of Portage, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 2, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-0551-2
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• Resumen

  • We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in o-minimal expansions of fields. Using it, we generalize the results in Fox et al. (J Eur Math Soc 19(6):1785–1810, 2017 ) on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalization of the Szemerédi–Trotter theorem.