Tuples are arbitrary functions $Fin ℓ → Fin n$, so repeated entries and every ordering of the entries are allowed. This is stronger than colouring the ℓ-element subsets of the ground set — that classical form is the special case where the colouring only depends on the increasing enumeration — and it is the form that applications consume, since the tuples they colour arise from arbitrary indexed families.

Homogeneity is stated as "tuples with entries in I and equal order types receive equal colours", rather than as the existence of a function from order types to colours through which the colouring factors. The two are equivalent — such a function is obtained by choice — and the stated form carries no choice and needs no default colour for the order types that no tuple over I realizes.

The ground set is $Fin n$ with its standard linear order, the canonical carrier of this submission; sizes are $Set.ncard$, stated as "at least". As in the other statements, no hypothesis is placed on the number of colours: with $k = 0$ there is no colouring at all.