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

The optimality of the Erdős–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 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 maximal triangle-free or odd-cycle-free colouring into $$\kappa$$ . We resolve the question positively for odd-cycle-free colourings.