Approximate Euclidean Ramsey theorems

• Main Author: Adrian Dumitrescu
• Format: Article
• Language: English
• Published: Carleton University 2011-04-01
• Series: Journal of Computational Geometry
• Online Access: http://jocg.org/index.php/jocg/article/view/38

According to a classical result of Szemerédi, every dense subset of 1,2,…,<em>N</em> contains an arbitrary long arithmetic progression, if <em>N</em> is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of {1,2,…,<em>N</em>}^<em>d</em> contains an arbitrary large grid, if <em>N</em> is large enough. Here we generalize these results for separated point sets on the line and respectively in the Euclidean space: (i) every dense separated set of points in some interval [0,<em>L</em>] on the line contains an arbitrary long approximate arithmetic progression, if <em>L</em> is large enough. (ii) every dense separated set of points in the <em>d</em>-dimensional cube [0,<em>L</em>]^<em>d</em> in R^<em>d</em> contains an arbitrary large approximate grid, if <em>L</em> is large enough. A further generalization for any finite pattern in R^<em>d</em> is also established. The separation condition is shown to be necessary for such results to hold. In the end we show that every sufficiently large point set in R^<em>d</em> contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.