  We present new results on approximate colourings of graphs and, more
generally, approximate H-colourings and promise constraint satisfaction
problems.
  First, we show NP-hardness of colouring $k$-colourable graphs with
$\binom{k}{\lfloor k/2\rfloor}-1$ colours for every $k\geq 4$. This improves
the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness
of colouring $k$-colourable graphs with $2k-1$ colours for $k\geq 3$, and the
result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring
$k$-colourable graphs with $2^{k^{1/3}}$ colours for sufficiently large $k$.
Thus, for $k\geq 4$, we improve from known linear/sub-exponential gaps to
exponential gaps.
  Second, we show that the topology of the box complex of H alone determines
whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite,
H-colourable) G. This formalises the topological intuition behind the result of
Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is
NP-hard for all (3-colourable, non-bipartite) G. We use this technique to
establish NP-hardness of H-colouring of G-colourable graphs for H that include
but go beyond $K_3$, including square-free graphs and circular cliques (leaving
$K_4$ and larger cliques open).
  Underlying all of our proofs is a very general observation that adjoint
functors give reductions between promise constraint satisfaction problems.
