3 problems

Let $T$ be a complete first-order theory. Say that $T$ has the stable Kim-forking property over models if, for every model $M\models T$ and every $B\supseteq M$, whenever…

Stable Kim-forking conjecture. If a\mathrel{\mathop{\mathpalette\setbox0=\text{x}\kern\wd0\text to 0pt{\mathchardef \nn=12854\hssnn\kern1.4\wd0\hss}\text to 0pt{\hssmid…

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…