## ON A GENERALIZATION OF SZEMERÉDI'S THEOREM 1

I. D. SHKREDOV a1
a1 Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia ishkredov@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.

(Published Online October 13 2006)