13 problems

Let $\lambda$ be a cardinal. For an abstract elementary class (AEC) $\boldsymbol{K}$, write $\operatorname{LS}(\boldsymbol{K})$ for its Löwenheim–Skolem number, and say that…

The paper considers two problems, called the infinite problem and the bad-clique problem, concerning the type-respecting amalgamation property. Negative-answer conjecture. The answ…

Baldwin's conjecture. No generic with a small theory will be strictly superstable.

Categoricity-above-the-Hanf-number conjecture. If $\mathcal{K}$ is categorical above the Hanf number, then $\mathcal{K}$ has the $\mu$-amalgamation property for every $\mu$.

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class, let $\operatorname{Hanf}(\operatorname{\mathcal{K}})$ be its Hanf number, and let $LS(\operatorname{\mathcal{K}})$…

Let $\operatorname{\mathcal{K}}$ be an abstract elementary class. Say that $\operatorname{\mathcal{K}}$ is categorical in a cardinal when it has exactly one model of that cardinali…

Let $T$ be a simple theory, let $K(n)$-simplicity denote the corresponding simplicity property, and let $(n+2)$-CA mean $(n+2)$-complete amalgamation over any set. For $n>2$, these…

In a simple theory, failure of 4-amalgamation is the property under discussion. Goodrick's conjecture. Failure of 4-amalgamation comes from some, in a suitable sense, interpretable…

EPPA–Ramsey expansion conjecture. Every strong amalgamation class with EPPA has a precompact Ramsey expansion.

A semilinear De Morgan monoid is a De Morgan monoid whose lattice reduct is a semilinear residuated lattice, and let $\Omega({{\sf SDMM}})$ denote the collection of varieties of se…

Let $T$ be a simple theory and let $3 \leq k \leq \aleph_0$. Consider the following conditions: (A) $T$ has $\mathcal{P}^-(k)$-amalgamation of models; (B) $T$ has $<k$-type amalgam…

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…

The full-class conjecture. One might optimistically conjecture that