  A linearly ordered (LO) $k$-colouring of a hypergraph assigns to each vertex
a colour from the set $\{0,1,\ldots,k-1\}$ in such a way that each hyperedge
has a unique maximum element. Barto, Batistelli, and Berg conjectured that it
is NP-hard to find an LO $k$-colouring of an LO 2-colourable 3-uniform
hypergraph for any constant $k\geq 2$ [STACS'21] but even the case $k=3$ is
still open. Nakajima and \v{Z}ivn\'{y} gave polynomial-time algorithms for
finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with
$O^*(\sqrt{n})$ colours [ICALP'22] and an LO colouring with $O^*(\sqrt[3]{n})$
colours [ACM ToCT'23]. Very recently, Louis, Newman, and Ray gave an SDP-based
algorithm with $O^*(\sqrt[5]{n})$ colours. We present two simple
polynomial-time algorithms that find an LO colouring with $O(\log_2(n))$
colours, which is an exponential improvement.
