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.
Categories: math.LO math.CO