Christoph Spiegel (Catalunya): Intervals in the Hales-Jewett Theorem

The Hales–Jewett Theorem states that any $r$–colouring of $[m]^n$ contains a monochromatic combinatorial line if $n$ is large enough. Shelah’s proof of the theorem implies that for $m = 3$ there always exists a monochromatic combinatorial lines whose set of active coordinates is the union of at most $r$ intervals. I will present some recent findings relating to this observation. This is joint work with Nina Kamcev.