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In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erdős 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. 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 Erdős, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.

We introduce the Bounded Axiom A Forcing Axiom (BA AFA). It turns out that it is equiconsistent with the existence of a regular Σ2-correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA).A plenitude of Forcing Axioms has been considered during the last thirty years. The class of forcing notions satisfying Axiom A has been somewhat neglected, though. We herewith want to draw attention to what we define to be the Bounded Axiom A Forcing Axiom.Notation 1 In this paper Σ cl names the class of all countably closed notions of forcing. The class of forcing notions satisfying the countable chain condition is denoted by C cc . By A A we mean the class of all forcing notions satisfying Axiom A, by P rop the class of all proper notions of forcing and by S sp the class of forcing notions preserving the stationarity of subsets of ℵ 1 .The class of all complete Boolean algebras is denoted by cba and ro(P) denotes the regular open algebra of a forcing notion P. X is the cardinality and ϕ"X the pointwise image of X under ϕ. Func refers to the class of all functions. For an ordinal α and a function f in a forcing extension byḟ (α) we mean a name for f (α).For the definition and the properties of the regular open algebra of a forcing notion see [7, Lemma II.3.3 and Section VII.7].Definition 2 A forcing notion P = P, 0 P satisfies Axiom A if and only if there exists a sequence n P | n ∈ ω \ 1 of partial orders on P such that the following hold:

We prove that for regular cardinals κ, combinations of the stick principle at κ and certain cardinal characteristics at κ being κ + causes the partition relations such as ω 1 −→ (ω 1 , ω + 2) 2 and (κ + ) 2 −→ (κ + κ, 4) 2 to fail. Polarised partition relations are also considered, and the results are used to answer several problems posed