TREE-INDISCERNIBILITY IN SOP$_1$ AND ANTICHAIN TREES (Model theoretic aspects of the notion of independence and dimension)

Abstract

We study some tree properties and related indiscernibilities. First, we show that there is a tree-indiscernibility which preserves witnesses of SOP1. Secondly we introduce notions of antichain tree property and show that every SOP1-NSOP2 theory (having SOP1 but not SOP2) has an antichain tree by using that tree-indiscernibility. And we construct a structure witnessing SOP1-NSOP2 in the formula level, i.e. there is a formula having SOP1 but any finite conjunction of it does not have SOP2. (This work is joint work with JinHoo Ahn at Yonsei University.)