  By a finite type-graph we mean a graph whose set of vertices is the set of
all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and
in which two such sets are adjacent if and only if they realise a certain order
type specified in advance. Examples of such graphs have been investigated in a
great variety of contexts in the literature with particular attention being
paid to their chromatic number. In recent joint work with Tomasz {\L}uczak, two
of the authors embarked on a systematic study of the chromatic numbers of such
type-graphs, formulated a general conjecture determining this number up to a
multiplicative factor, and proved various results of this kind. In this article
we fully prove this conjecture.
