Notions around the Tree Property 1

In this paper, we study the notions related to the tree property 1 (=TP1) or equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k-TP1. Then we introduce the notions of weak k-TP1 for k ≥ 2, and also supply type-counting criteria for those. We do not know whether weak k-TP1 implies TP1, but at least we prove that each weak k-TP1 implies SOP1. Our generalization of tree-indiscernibility results in [5] is crucially used throughout the paper. As is well-known, a complete theory T is simple if and only if it does not have the tree property. A theory being simple is characterized by having an (automorphism-invariant) independence relation satisfying symmetry, transitivity, extension (i.e. for any c and A ⊆ B, there is c(≡A c) such that c′ is independent with B over A), local character, finite character, anti-reflexivity (a tuple c is always dependent with itself over any set B unless c ∈ acl(B)), and type-amalgamation over a model [10]. But still, it is natural to ask whether there is a suitable class of theories (possibly properly containing that of simple theories) having an independence relation satisfying fewer number of independence axioms aforementioned. Indeed the class of rosy theories is characterized by having an independence relation forM satisfying all the axioms except for type-amalgamation over a model. Thus all simple and o-minimal theories are rosy [6],[1]. On the other hand, there are natural examples (which need not be rosy but) having an independence relation for M satisfying all the mentioned axioms including stationarity over a model (which implies type-amalgamation over a model), except for local character. In [2], such theories are called mock stable or mock simple, respectively.