On some Fraïssé classes with free amalgamation
Abstract: In this talk I will present the class of NSOP1 theories and some examples. This class was studied recently by Kim, Ramsey, Chernikov, Kaplan, Dobrowolsky, D'Elbée and Kruckmann, and is seen as a generalization of simple theories. The main tools we have in this context are independence relations, for example non Kim-forking independence, which is the equivalent of non-forking in simple theories, and which has a Kim-Pillay criterion. I will briefly present these tools and some of the main results. Most of the examples we have of strictly NSOP1 theories (meaning NSOP1 non-simple) are built from simple theories by adding some generic structure, and I will present some examples of this type. We begin with some assumption on a Fraïssé class, their consequences on the theory of the limit of this class (in particular the fact that it is NSOP1) using mostly independence relations, and then see how we can modify this class in order to make the limit non simple.
