Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T04:08:31.654Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A GENERALIZATION OF SZEMERÉDI'S THEOREM

Published online by Cambridge University Press:  13 October 2006

I. D. SHKREDOV
I. D. SHKREDOV
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119992, [email protected], [email protected]
Get access

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

35J2537A15
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 93 , Issue 3 , November 2006 , pp. 723 - 760
Copyright
2006 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).