18 problems

Let $G$ be a closed oligomorphic permutation group, meaning a closed permutation group with finitely many orbits on $n$-tuples for every $n\geq 1$. An expansion of an $\omega$-cate…

Primitive intermediate-growth conjecture. If a primitive $c9$-categorical structure $c4$ satisfies

Slowest-growth conjecture. If an $c9$-categorical structure $c4$ is not interpretable in $(\mathbb Q;<)$, then

Let $\mathfrak{B}$ be a first-order reduct of a finitely bounded homogeneous structure in a finite relational signature. The constraint satisfaction problem…

Let $R$ be a ring that is both omega-categorical, meaning that its theory has a unique countable model up to isomorphism, and pseudofinite, meaning that it is infinite and every fi…

Let $G$ be a countably infinite group that is both omega-categorical, meaning that its theory has a unique countable model up to isomorphism, and pseudofinite, meaning that it is i…

Tractable-or-NP-hard dichotomy conjecture. If either

Simon's near-$2^n$ conjecture. Then $G$ is either $\operatorname{Aut}(S(2))$, or the group of automorphisms and anti-automorphisms of $S(2)$.

Generalized Fibonacci growth conjecture. If

Monadic NIP growth conjecture.

An omega-categorical structure is a structure whose theory has a unique countable model up to isomorphism. A group or ring is tame or supertame in the sense used in the paper's mod…

Let $M$ be an $\omega$-categorical structure. Its Lascar group is the quotient of $\operatorname{Aut}(M)$ by the subgroup of Lascar strong automorphisms. Expansion conjecture. Any…

An $c9$-categorical structure is a structure whose theory has, up to isomorphism, a unique countable model. A structure admits a CIR when it has the property called CIR in the pape…

Homomorphic-equivalence CSP dichotomy conjecture. One of the following holds:

Model-complete-core CSP dichotomy conjecture. One of the following holds:

Let $\Gamma$ be an $\omega$-categorical structure with finite relational signature. A family of totally symmetric polymorphisms consists of polymorphisms of every arity whose value i…

NIP conjecture for omega-categorical rings. Every $\omega$-categorical ring with NIP is nilpotent-by-finite.

NIP conjecture for omega-categorical groups. Every $\omega$-categorical group with NIP is nilpotent-by-finite.