论文标题
不可数的图形的极端三角形和无循环的色
Extremal triangle-free and odd-cycle-free colourings of uncountable graphs
论文作者
论文摘要
$Δ_κ$δ_κ见证了erdős-rado定理的最佳性：[2^κ]^2 \rightarrowκ$记录两个函数之间分歧的最小分。这种着色没有单色三角形或更一般的奇数循环。我们研究了许多问题，这些问题调查了$Δ_κ$是\ emph {xudyal}这种无三角形或无循环的着色的程度。首先，我们引入$δ$ - 退缩和几乎$δ$ - 回归彩色的概念，并研究必须以单色子图显示的结构。我们还考虑了一个问题，即$δ_κ$是否具有任何\ emph {maximal}的最小基数，无三角形或无循环的颜色为$κ$。我们为无奇数的色彩积极解决问题。
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.
