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

Progressions in Euclidean Ramsey theory

Conlon and Wu showed that there is a red/blue-coloring of $\mathbb{E}^n$ that does not contain $3$ red collinear points separated by unit distance and $m=10^{50}$ blue collinear points separated by unit distance. We prove that the statement holds with $m=1177$. We show similar results with different distances between the points

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

Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory

We show that any finite affinely independent set can be isometrically embedded into a regular polygonal torus, that is, a finite product of regular polygons. As a consequence, with a straightforward application of K\v{r}\'{i}\v{z}'s theorem, we get an alternative proof of the fact that all finite affinely independent sets are Ramsey, a result which was originally proved by Frankl and R\"{o}dl.Comment: 7 pages; corrected typo

Triangles, Triangles and, Yes, More Triangles: Explorations in Euclidean Ramsey Theory

Several important general theorems of Euclidean Ramsey Theory are presented with an emphasis on trying to prove or disprove the 1973 conjecture of Erdös et al. that for all triangles, except for equilateral triangles, it is possible to find a monochromatic coloring of the vertices in any two colorings of the plane. Further investigation included looking at triangles in greater dimensions

A recursive Lov\'asz theta number for simplex-avoiding sets

We recursively extend the Lov\'asz theta number to geometric hypergraphs on the unit sphere and on Euclidean space, obtaining an upper bound for the independence ratio of these hypergraphs. As an application we reprove a result in Euclidean Ramsey theory in the measurable setting, namely that every $k$-simplex is exponentially Ramsey, and we improve existing bounds for the base of the exponential.Comment: 13 pages, 3 figure