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 Δ κ : [ 2 κ ] 2 → κ 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 Δ κ is an extremal such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of Δ -regressive and almost Δ -regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether Δ κ has the minimal cardinality of any maximal triangle-free or odd-cycle-free colouring into κ . We resolve the question positively for odd-cycle-free colourings. [ABSTRACT FROM AUTHOR]