49 problems

Let $\mathfrak{A}$ be a countable homogeneous structure in a finite relational signature. A reduct of $\mathfrak{A}$ is a structure obtained by restricting its definable relations,…

Let $\Gamma$ be a homogeneous structure with a finite relational signature. An expansion of $\Gamma$ is a structure obtained by adding relations to its signature; its age is the cl…

Let $T$ be the theory of an expansion of an infinite group. A theory $T$ is $(m,n)$-homogeneous if, whenever ${\mathscr{M}}\models T$ and…

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

The Primitivity Conjecture. Modulo the agreement of certain orders up to reversal, every primitive homogeneous finite-dimensional permutation structure is the Fraïssé limit of all…

Lachlan class characterization conjecture. Lachlan's class equals precisely the class of structures that are interdefinable with the model-complete core of a structure interpretabl…

A relational structure $A$ is finitely bounded homogeneous if it is homogeneous and its age is characterized by finitely many forbidden finite structures. Let $B$ be a relational s…

Let $\mathbb A$ be a first-order reduct of a finitely bounded homogeneous structure. The constraint satisfaction problem of $\mathbb A$ is denoted by…

The reduct dichotomy conjecture. CSPs of reducts of finitely bounded homogeneous structures are in $\operatorname{\mathbf{P}}$ or $\operatorname{\mathbf{NP}}$-complete. Equivalentl…

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

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…

The tractability conjecture. If ${\mathbb A}$ does not pp-construct ${\mathbb K}_3$, then $\operatorname{CSP}({\mathbb A})$ is polynomial-time solvable.

Bodirsky's Ramsey expansion conjecture. Every finitely bounded homogeneous structure has a finitely bounded Ramsey expansion.

Let ${\mathfrak{B}}$ be a reduct of a finitely bounded homogeneous structure. Say that a structure is primitively positively constructible in another structure if it is homomorphic…

Let ${\mathfrak{B}}$ be a structure, and suppose that $\operatorname{Aut}({\mathfrak{B}})$ contains the automorphism group of a relational structure ${\mathfrak{A}}$. Assume that…

Homogeneous-structure conjecture. (i) Let $M$ be a homogeneous structure over a finite relational language $L$. Then there is an m.e.c. with ultraproduct elementarily equivalent to…

Pp-construction conjecture. If $\operatorname{PCSP}(G,H)$ is NP-hard, then $K_3$ has a pp-construction in $S$.

Infinite-domain tractability conjecture. Such a structure has a polynomial-time tractable CSP unless it admits a primitive positive interpretation of a structure homomorphically eq…

The stability conjecture. Any infinite set-homogeneous $k$-hypergraph with stable theory is complete or has complete complement.

Finiteness conjecture. There are only finitely many first-order reducts of $\mathcal{A}$ up to interdefinability.

Classification conjecture. Trace-equivalence classes of finitely homogeneous structures should correspond to natural model-theoretic properties.

Simon's conjecture. There are only countably many finitely homogeneous NIP structures up to isointerdefinibility.

Countability conjecture. There are $\aleph_0$ finitely homogeneous structures up to trace equivalence.

Strong conjecture. For every $k$ there are only finitely many finitely homogeneous structures in a $k$-ary language up to trace equivalence.

Let $\str{H}$ be the countable, $\omega$-categorical structure constructed using the Hrushovski predimension construction, and let $\Gamma$ be the amenable subgroup of…