Approximate euclidean ramsey theorems

According to a classical result of Szemerédi, every dense subset of 1, 2, . . . , N contains an arbitrary long arithmetic progression, if N is large enough. Its analogue in higher dimensions due to Fürstenberg and Katznelson says that every dense subset of {1, 2, . . . , N}d contains an arbitrary large grid, if N 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, L] on the line contains an arbitrary long approximate arithmetic progression, if L is large enough. (ii) every dense separated set of points in the d-dimensional cube [0, L] in R contains an arbitrary large approximate grid, if L is large enough. A further generalization for any finite pattern in R 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 contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.