Canonical partition theorems for parameter sets (Q798658)

For every partition relation with bounded sets of colours it is possible to generalize this relation to a related canonical partition relation with an unbounded set of colours. In the first case one is asking if there exist special monochromatic substructures while in the second case one is interested in substructures who's colours depend on some well defined items. So every partition problem has an analogous canonical partition problem. In this paper the authors proved the canonical version of the Hales-Jewett partition result. Then they showed that some of well- known results of canonical partition relations are corollaries of the canonical version of the Hales-Jewett result. That's not astonishing because it reflects the strongness of the Hales-Jewett theorem.