The optimality of the Erdős–Rado theorem for pairs is witnessed by the colouring $$\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$$ 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 $$\Delta_\kappa$$ Δ κ is an extremal such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $$\Delta$$ Δ -regressive and almost $$\Delta$$ Δ -regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $$\Delta_\kappa$$ Δ κ has the minimal cardinality of any maximal triangle-free or odd-cycle-free colouring into $$\kappa$$ κ . We resolve the question positively for odd-cycle-free colourings.

中文翻译：

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不可数图的极值无三角形和无奇数循环着色

Erdős–Rado 定理对的最优性由着色 $$\Delta_\kappa 见证： [2^\kappa]^2 \rightarrow \kappa$$ Δ κ : [ 2 κ ] 2 → κ 记录最小点两个函数之间的分歧。这种颜色没有单色三角形，或者更一般地说，没有奇数周期。我们调查了许多问题，调查 $$\Delta_\kappa$$ Δ κ 在多大程度上是这样的无三角形或无奇数循环着色的极值。我们首先介绍 $$\Delta$$ Δ 回归和几乎 $$\Delta$$ Δ 回归着色的概念，并研究必须作为此类着色的单色子图出现的结构。我们还考虑了关于 $$\Delta_\kappa$$ Δ κ 是否具有任何最大无三角形或无奇数循环着色到 $$\kappa$$ κ 的最小基数的问题。