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ON A GENERALIZATION OF SZEMERÉDI'S THEOREM

Published online by Cambridge University Press:  13 October 2006

I. D. SHKREDOV
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119992, Russiaishkredov@rambler.ru, ishkredo@mech.math.msu.su
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Abstract

Let $N$ be a natural number and $A \subseteq [1, \dots, N]^2$ be a set of cardinality at least $N^2 / (\log \log N)^c$, where $c > 0$ is an absolute constant. We prove that $A$ contains a triple $\{(k, m), (k+d, m), (k, m+d) \}$, where $d > 0$. This theorem is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.

Keywords

Type
Research Article
Copyright
2006 London Mathematical Society

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Footnotes

This work was supported by the program ‘Leading Scientific Schools’ (project no. 136.2003.1), and by RFFI grant no. 02-01-00912 and INTAS (grant no. 03-51-5-70).