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 $i,j\leq k$.

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