Choiceless Ramsey Theory of Linear Orders

Together with the folklore result (using AC) that ψ 6 −→(ω, ω∗)2 this puts things into perspective. It is known that one can have very strong partition properties in models of ZF violating AC, consider for example Mathias’s result that ω −→ (ω)2 is consistent with ZF—cf. [6] or Martin’s discovery that AD implies ω1 −→ (ω1)1 which failed to be published by him (but cf. [2, 3, 4, 5]). We focus on linear orders of the form 〈2, <lex〉 for ordinals α and prove positive and negative partition relations, an example of the former is the following theorem.