Dividing lines between positive theories

We generalise the properties OP, IP, k-TP, TP₁, k-TP₂, SOP₁, SOP₂ and SOP₃ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having TP and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has OP iff it has IP or SOP₁ and that T has TP iff it has SOP₁ or TP₂, analogous to the well-known results in full first-order logic where SOP₁ is replaced by SOP in the former and by TP₁ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.