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.Comment: 11 pages, 1 figure

Remarks on monochromatic configurations for finite colorings of the plane

Gurevich had conjectured that for any finite coloring of the Euclidean plane, there always exists a triangle of unit area with monochromatic vertices. Graham ([5], [6]) gave the first proof of this conjecture; a much shorter proof has been obtained recently by Dumitrescu and Jiang [4]. A similar result in the case of a trapezium, claimed by the present authors in [3] does not hold due to an error and a weaker result is recovered for quadrilaterals in this paper. We also take up the original question of triangle