24 problems

Bodirsky–Pinsker conjecture. $\operatorname{CSP}(\mathbb C)$ is either in P or NP-complete.

Kaplan–Simon conjecture. Every $c9$-categorical structure admits an $c9$-categorical expansion with a CIR.

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…

Monadic NIP–exponential growth–wqo age conjecture. The following conditions are equivalent: $M$ is monadically NIP; the sequence $\bigl(f_k(\operatorname{Aut}(M))\bigr)$ is bounded…

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.

Assume that $M$ is an $c9$-categorical NIP structure satisfying the additional rank-theoretic condition of rosiness described in the paper. The conjecture compares such structures…

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…

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

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

Abelian-by-finite conjecture. Every stable $\omega$-categorical group is abelian-by-finite.

Let $\Gamma$ be the model-complete core of a structure first-order definable in the Fraïssé limit of a finitely bounded Fraïssé class. An expansion $\Gamma'$ of $\Gamma$ by finitel…

Let $G$ be an $c9$-categorical group with NIP, meaning that its first-order theory has the non-independence property. A group is nilpotent-by-finite if it has a nilpotent subgroup…

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…

Let $Gamma$ be an $omega$-categorical relational structure. A semi-lattice polymorphism is a commutative, associative and idempotent binary polymorphism. The semi-lattice polymorph…

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.