Monochromatic Hilbert Cubes and Arithmetic Progressions

  title={Monochromatic Hilbert Cubes and Arithmetic Progressions},
  author={J{\'o}zsef Balogh and Mikhail Lavrov and George Shakan and Adam Zsolt Wagner},
  journal={Electron. J. Comb.},
The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic affine $k$–cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the… Expand


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