I. Shkredov

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.

2000 Mathematics Subject Classification: 35J25, 37A15.

E-mail:
ishkredov@rambler.ru
ishkredo@mech.math.msu.su