Mathias Beiglböck: Topological methods in Ramsey theory

The celebrated Theorem of van der Waerden is very simple to state as well as non trivial to prove: If $ \mathbb{N}$ is finitely coloured, there exists a monochrome set $ A\subseteq \mathbb{N}$ which contains arbitrarily long arithmetic progressions. The corresponding statement about geometric progressions is also known to be true. We use the algebraic structure of the Stone-Cech-Compactification of $ \mathbb{N}$ to derive common extensions of these results: For example we show that for $ k \in \mathbb{N}$ there exist $ a,d,r\in A$ such that $ (a+di)r^j\in A$ for all $ i,j\leq k$.

Back to the Index

Please send comments and corrections to Thomas Klausner.