# Partitioning subsets of generalised scattered orders

## TL;DR

This paper explores the partition relations of generalized scattered linear orders, extending classical results and providing new insights into their combinatorial properties for uncountable cardinals.

## Contribution

It introduces new partition relation results for classes of $oldsymbol{ ext{kappa}}$-scattered linear orders, generalizing previous work and analogues of the Milner-Rado paradox.

## Key findings

- Failure of certain partition relations under specific set-theoretic assumptions.
- Extension of partition results to uncountable $oldsymbol{ ext{kappa}}$-scattered linear orders.
- Connections to classical and modern research in ordered set theory.

## 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.