Lines in Euclidean Ramsey theory

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erd\H{o}s, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.Comment: 7 page

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

Recent trends in Euclidean Ramsey theory

AbstractWe give a brief summary of several new results in Euclidean Ramsey theory, a subject which typically investigates properties of configurations in Euclidean space which are preserved under finite partitions of the space

Euclidean ramsey theorems. I

AbstractThe general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B. For a ϵ A denote by R(a) the set {b ϵ B | (a, b) ϵ R}. R is called r-Ramsey if for any r-part partition of B there is some a ϵ A with R(a) in one part. We investigate questions of whether or not certain R are r-Ramsey where B is a Euclidean space and R is defined geometrically