A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023).

翻译：超图的线性序$k$-着色是指用颜色$1,\dots,k$对顶点进行着色，使得每条边包含唯一的最大颜色。判定输入超图是否具有固定颜色数的线性序$k$-着色是NP完全的（在图的特例中，线性序着色与通常的图着色一致）。本文研究超图"线性序色数"近似计算的复杂性。我们证明以下承诺问题是NP完全的：给定一个3一致超图，区分其是否为线性序3-可着色，以及其甚至不是线性序4-可着色两种情况。该结果通过代数、拓扑和组合方法的结合得到证明，构建并扩展了Krokhin、Opršal、Wrochna和Živný（2023）引入的用于研究近似图着色的拓扑方法。