  The optimality of the Erd\H{o}s-Rado theorem for pairs is witnessed by the
colouring $\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$ recording the least
point of disagreement between two functions. This colouring has no
monochromatic triangles or, more generally, odd cycles. We investigate a number
of questions investigating the extent to which $\Delta_\kappa$ is an
\emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by
introducing the notion of $\Delta$-regressive and almost $\Delta$-regressive
colourings and studying the structures that must appear as monochromatic
subgraphs for such colourings. We also consider the question as to whether
$\Delta_\kappa$ has the minimal cardinality of any \emph{maximal} triangle-free
or odd-cycle-free colouring into $\kappa$. We resolve the question positively
for odd-cycle-free colourings.
