Colouring of a set ($c:X\to C$)

A colouring assigns to each member of a set $X$ a colour in another set $C$. Its nonempty fibres form a partition of a set. The colour set may be infinite; a finite colouring is the special case with finitely many colours. A homogeneous set for a colouring of finite subsets has all its subsets of the specified size assigned one colour.