The chromatic number of finite type-graphs

By a finite type-graph we mean a graph whose set of vertices is the set of all k-subsets of [n]={1,2,…,n} for some integers n≥k≥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 Łuczak, 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.