Mervyn Tong (Leeds)

On stable and distal expansions of $(\mathbb{Z}, +)$ and $(\mathbb{Z}, <, +)$ by a unary predicate

Proper expansions of $(\mathbb{Z}, +)$ (respectively $(\mathbb{Z}, <, +)$) by a unary predicate have proven to be a rich source of interesting stable (respectively distal) structures. Precisely which predicates give a stable (respectively distal) expansion remains an open question that, not only is of independent interest, but also has the potential to provide counterexamples to open problems on model-theoretic dividing lines. In this workshop, we survey some results that provide a partial answer to this question and discuss strategies to tackle this and adjacent open problems. In particular, we will discuss the existence of non-distal NIP expansions of $(\mathbb{N}, +)$, and we will consider a candidate counterexample to the conjecture that a structure has a distal expansion if and only if every formula has the strong Erdős-Hajnal property.