The chromatic number of finite type-graphs

TL;DR

This paper proves a conjecture about the chromatic number of finite type-graphs, which are graphs defined on k-subsets of [n] with edges determined by specific order types, advancing understanding in graph coloring.

Contribution

The paper fully proves a previously formulated conjecture on the chromatic number of finite type-graphs, providing a complete theoretical understanding.

Findings

Confirmed the conjecture for all finite type-graphs

Established the exact asymptotic behavior of the chromatic number

Unified previous partial results into a comprehensive proof

Abstract

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.