Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erd\H{o}s-Rado theorem for pairs is witnessed by the colouring ${\mathrm{\Delta }}_{\kappa }:[2^{\kappa }{]}^2\to \kappa$ 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 ${\mathrm{\Delta }}_{\kappa }$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $\mathrm{\Delta }$-regressive and almost $\mathrm{\Delta }$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether ${\mathrm{\Delta }}_{\kappa }$ 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.