25 problems

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

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.

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 $S$.

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

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

Modulus-of-continuity conjecture. Every group topology on $G$ strictly coarser than $\tau_{st}$ is of the form $\tau_f$ for some $\mathcal{R}$-modulus of continuity.

Primitive homogeneous structures conjecture. The distinguishing number of every primitive homogeneous countably infinite structure is two or infinite.

Let $M$ be a finitely homogeneous NIP structure. Structural conjecture. The following hold: 1. The automorphism group $\operatorname{Aut}(M)$ acts oligomorphically on the space of…

Construction completeness conjecture. This construction produces all homogeneous finite-dimensional permutation structures.

Homogeneity-after-naming conjecture. The class $C_2$ remains homogeneous after naming $C_1$.

3-type extension conjecture. All 3-types involving realized 2-types are realized in $\Gamma$.

3-type realization conjecture. If $\Gamma$ realizes all 3-types, then $\Gamma$ is fully generic.

Primitivity Conjecture. Every primitive homogeneous finite-dimensional permutation structure can be constructed as follows: identify certain orders, up to reversal; then take the F…

Classification conjecture. Every homogeneous finite-dimensional permutation structure with lattice of $\emptyset$-definable equivalence relations isomorphic to $\Lambda$ is interde…

Genericity conjecture. Then $93$ is fully generic.

Braun's Primitivity Conjecture. Every primitive homogeneous finite-dimensional permutation structure can be constructed by this procedure, yielding a fully generic structure, possi…

Cherlin's metric homogeneity classification conjecture. The countable metrically homogeneous graphs are the following: those of diameter $\delta\leq 2$, namely the homogeneous grap…

A structure is primitive if no nontrivial equivalence relation on its domain is invariant under its automorphism group; it is binary if its language has relation symbols of arity a…

Dichotomy conjecture. $\operatorname{CSP}(\Gamma)$ is either in P or NP-complete.

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…

Thomas's conjecture. The structure $\Delta$ has only finitely many reducts up to first-order interdefinability.