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Partitioning subsets of generalised scattered orders. (English) Zbl 07056563

Summary: In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, P. Erdős and R. Rado founded the partition calculus in a seminal paper [Bull. Am. Math. Soc. 62, 427–489 (1956; Zbl 0071.05105)]. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. We consider analogues for these order-types of known partition theorems for ordinals or scattered orders and prove a partition theorem from assumptions about cardinal characteristics. Together, this continues older research by Erdös, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.

MSC:

03E02 Partition relations
03E17 Cardinal characteristics of the continuum
05C63 Infinite graphs
05D10 Ramsey theory
06A05 Total orders

Citations:

Zbl 0071.05105
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