Partitioning subsets of generalised scattered orders

Abstract

In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erd\H{o}s and Rado founded the partition calculus in a seminal paper. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. It provides analogues of the Milner-Rado paradox for these orders instead of ordinals. For infinite, regular $\kappa$, we investigate the extent to which the classes of $\kappa$-scattered, weakly $\kappa$-scattered, and $\kappa$-saturated linear orders of size $\kappa$ are closed under the partition relation $\tau \rightarrow (\varphi, n)^2$ for all $n < \omega$. We prove that for a regular cardinal $\kappa$ such that the stick principle holds at $\kappa$ and $\mathfrak{b}_\kappa = \kappa^+$, the partition relation $\kappa^+\kappa \rightarrow (\kappa^+\kappa, 3)^2$ fails. Finally we generalise a result of Komj\'{a}th and Shelah about partitions of scattered linear orders to a similar result about partitions of $\kappa$-scattered linear orders for uncountable $\kappa$. Together this continues older research by Erd\H{o}s, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, D\v{z}amonja, Komj\'{a}th, Shelah and Thompson.