Title: Enriching a predicate and tame expansions of the integers


Abstract: Given a structure M and a stably embedded 0-definable set Q, we prove tameness preservation results when enriching the induced structure on Q by some further structure. 

In particular, we show that if the original structure and the enriched structure on Q are stable (resp., superstable, ω​​-stable), then so is the induced enrichment of M. 

Using these results  we construct the first known examples of strictly stable expansions of (ℤ,+)​​. More generally, we show that any stable (resp., superstable) countable graph can be defined in a stable (resp., superstable) expansion of (ℤ,+)​​ by some unary predicate A⊆ℕ​​. 

Joint work with G. Conant, C. d'Elbee, L. Jimenez and S. Rideau-Kikuchi