Colouring versus density in integers and Hales-Jewett cubes

Christian Reiher, Vojtěch Rödl, Marcelo Sales

We construct for every integer $k\ge 3$ and every real $\mu \in (0,\frac{k-1}{k})$ a set of integers $X=X(k,\mu )$ which, when coloured with finitely many colours, contains a monochrom…