9 problems

A structure is monadically $IP$ if every expansion by unary predicates is NIP. Monadic NIP trace-definability conjecture. Monadically NIP structures cannot trace define infinite gr…

Finite-arity conjecture. For every $d\in\mathbb{N}_{\ge 1}$, there are only finitely many $d$-ary finitely homogeneous structures modulo trace equivalence.

NIP group conjecture. If $\mathscr{M}$ is $\aleph_0$-categorical and NIP, then $\mathscr{M}$ interprets an infinite group if and only if $\mathscr{M}$ trace defines an infinite gro…

Binary trace-equivalence conjecture. Every structure admitting quantifier elimination in a finite binary relational language is trace equivalent to either the trivial structure,…

Chernikov3Hempel conjecture. Every IP field is $k$-IP for all $k$; equivalently, every IP field is locally trace maximal.

Local trace maximality conjecture. The theory $T$ $\infty$-trace defines $D_\infty(T)$ if and only if $T$ is locally trace maximal if and only if $D_\infty(T)$ is trace maximal.

Countability and finite arity conjecture. There are only countably many finitely homogeneous structures modulo trace equivalence. More strongly, for every $k65 1$ there are only fi…

Ordered-abelian-group conjecture. An ordered abelian group cannot trace define an infinite field.

Irrational-slope trace-definability conjecture. A divisible ordered abelian group cannot trace define