10 problems

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

The maximality conjecture. The relation \mathrel{\raise0.2ex\text{\ooalign{\hidewidthvert\hidewidth\cr\raise-0.9ex\text{smile}}}}^{nm} is the maximum strongly admissible geom…

Conant-independence conjecture. Over models, Conant-independence coincides with \mathop{\mathpalette\setbox0=\text{x}\kern\wd0\text to 0pt{\hssmidhss} lower.9ht0 to 0pt{hss\s…

Nonexistence conjecture. There is no such independence relation defined over models of $T$.

Let $T$ be a complete theory with monster model $\mathbb{M}$. A set $e\subseteq\mathbb{M}$ is an extension basis if $a\mathrel{\smile^{f}_{e}}e$ for every $a\in\mathbb{M}$; $T$ has…

Let ${\mathbf{ELAF}}_{\lhd}$ be the category of ELA-fields with strong embeddings, where no very full kernel assumption is imposed, and let $\mathop{\mathpalette\Ind{}}^{\lhd}$ den…

Let $(X,T)$ be a distal minimal topological dynamical system, and let ${\bf Ind}^{[d]}_{ap}(X,T)$ denote its arithmetic-progression independence relation of order $d$. Let…

Let $T$ be a simple theory, and let $\thind$ denote thorn-independence in $T$. Canonicity conjecture. In any simple theory, $\thind$ is canonical. This conjecture proposes that tho…

Extension conjecture for lovely pairs. There exists an independence relation mathop{\mathpalette\Ind{}^{\!\!\!\!\rlap{scriptscriptstyletextnormal{2}}\,\,\,\,}} on…

Finitarity and definability conjecture. $\operatorname{cl}$ is always finitary. Moreover, $\operatorname{cl}$ is definable: for every small $A\subseteq\mathfrak C$, the set…