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 $Δ_κ: [2^κ]^2 \rightarrow κ$ 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 \emph{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 \emph{maximal} triangle-free or odd-cycle-free colouring into $κ$. We resolve the question positively for odd-cycle-free colourings.

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