## Abstract

We show that every hypersurface in ℝ^{s} × ℝ^{s} contains a large grid, i.e., the set of the form *S* × *T*, with *S, T* ⊂ ℝ^{s}. 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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## 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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### Cite this article

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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### Keywords

- Spectral Sequence
- Generic Polynomial
- Cohomology Ring
- Euler Class
- Algebraic Construction