On a generalization of Szemeredi's theorem

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. © 2006 London Mathematical Society.