Some Ternary Versions of Stability

Stability is the prototypical model theoretic dividing line. One interpretation is that a binary relation is stable if it is "close to unary": if the question $(x,y)\in E$ can be answered, at least most of the time, by knowing enough information about $x$, and separately enough information about $y$.

One natural question is asking how this can generalize to ternary (and higher-arity) relations. The connection to hypergraph regularity suggests an approach to identifying ternary stable-like properties, and also that there should be several versions, since a ternary relation could be almost unary, or almost binary, or a combination of these properties.

In this talk, I'll survey some of what we know about several of these "stable-like" ternary notions.