Approximate Euclidean Ramsey theorems

According to a classical result of Szemer\'{e}di, 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 F\"urstenberg 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 $[0,L]^d$ in $\RR^d$ contains an arbitrary large approximate grid, if $L$ is large enough. A further generalization for any finite pattern in $\RR^d$ 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 $\RR^d$ contains an arbitrarily large subset of almost collinear points. No separation condition is needed in this case.