Linearly Ordered Colourings of Hypergraphs

A linearly ordered (LO) $k$-colouring of an $r$-uniform hypergraph assigns integer from $\{1, \ldots, k \}$ to every vertex so that, in edge, the (multi)set colours has a unique maximum. Equivalently, for $r=3$, if two vertices edge are assigned same colour, then third is larger colour (as opposed different as classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on $3$-uniform hypergraphs context promise constraint satisfaction problems (PCSPs). We show results. First, given 3-uniform that admits $2$-colouring, one can find polynomial time with $k=O(\sqrt[3]{n \log n / n})$. Second, we establish NP-hardness finding constant uniformity $r\geq k+2$. In fact, determine relationships between polymorphism minions all uniformities 3$, which reveals key difference $r<k+2$ k+2$ may be independent interest. Using algebraic approach PCSPs, actually more general result establishing $\ell$-colourable $2 \leq \ell k$ $r \geq - + 4$.