Approximate Euclidean Ramsey theorems

TL;DR

This paper extends classical Euclidean Ramsey theorems to approximate structures in separated point sets within Euclidean spaces, establishing conditions under which large approximate arithmetic progressions, grids, and patterns must exist.

Contribution

It generalizes Euclidean Ramsey results to approximate configurations in separated point sets and introduces new conditions for their existence.

Findings

Dense separated sets contain large approximate progressions and grids

Separation condition is necessary for these results

Large point sets contain almost collinear subsets without separation constraints

Abstract

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.