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Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions

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Abstract

We show that every hypersurface in ℝs × ℝs contains a large grid, i.e., the set of the form S × T, with S, T ⊂ ℝs. We use this to deduce that the known constructions of extremal K 2,2-free and K 3,3-free graphs cannot be generalized to a similar construction of K s,s -free graphs for any s ≥ 4. We also give new constructions of extremal K s,t -free graphs for large t.

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Authors and Affiliations

  1. Mathematički Institut SANU, Knez Michailova 36, 11001, Beograd, Serbia

    Pavle V. M. Blagojević

  2. Institut für Mathematik Freie, Universitt Berlin, Arnimallee 2, 14195, Berlin, Germany

    Pavle V. M. Blagojević

  3. Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall 6113, Pittsburgh, PA, 15213-3890, USA

    Boris Bukh

  4. Department of Mathematics, Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia, 141700

    Roman Karasev

  5. Laboratory of Discrete and Computational Geometry, Yaroslavl’ State University, Sovetskaya st. 14, Yaroslavl’, Russia, 150000

    Roman Karasev

Authors
  1. Pavle V. M. Blagojević
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  2. Boris Bukh
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  3. Roman Karasev
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Corresponding author

Correspondence to Pavle V. M. Blagojević.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 247029-SDModels. Also supported by the grant ON 174008 of the Serbian Ministry of Education and Science.

Research was supported by University of Cambridge and by Churchill College.

Supported by the Dynasty Foundation, the President’s of Russian Federation grant MD-352.2012.1, the Russian Foundation for Basic Research grants 10-01-00096 and 10-01-00139, the Federal Program “Scientific and scientific-pedagogical staff of innovative Russia” 2009–2013, and the Russian government project 11.G34.31.0053.

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Blagojević, P.V.M., Bukh, B. & Karasev, R. Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions. Isr. J. Math. 197, 199–214 (2013). https://doi.org/10.1007/s11856-012-0184-z

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  • DOI: https://doi.org/10.1007/s11856-012-0184-z

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