14 problems

Assume that $T$ is a first-order theory and that … with $2^\lambda=\lambda^+$. Let $\langle M_\alpha:\alpha<\lambda^+\rangle$ be a $\prec$-increasing continuous sequence with…

Generic pair conjecture. The theory $T$ is independent if and only if

No strictly $n$-dependent fields conjecture. There are no strictly $n$-dependent fields for $n\geq 2$.

Field-dependence conjecture. Every $n$-dependent field is dependent.

Nonexistence conjecture. There are no strictly $k$-dependent fields, for any $k\geq 2$, whose theory is simple.

Fix a dependent diagram $D$ and a strongly compact cardinal $\theta>|T|$. Let $\theta<\lambda=\lambda^{<\lambda}$ and $\lambda^{+}=2^{\lambda}$. Let … be an increasing continuous s…

Let $\lambda$ be a cardinal such that … Let $T$ be a complete first-order theory, and let $M$ be a saturated model of $T$ of size $\lambda^{+}$. Write … where each $M_{\alpha}$ is…

Let $T$ be dependent, let $|T|<\lambda=\lambda^{<\lambda}$, let $\lambda^+=2^\lambda$, and let $\sigma=\operatorname{cf}(\sigma)<\lambda$. Suppose…

Let $T$ be a first-order theory, let $\lambda=\lambda^{<\lambda}>|T|$ with $2^\lambda=\lambda^+$, and let $\langle M_\alpha:\alpha<\lambda^+\rangle$ be a continuously increasing el…

Let $T$ be a first-order theory, let $M$ be a model of $T$, and let $\kappa$ be an infinite cardinal. A sequence $\langle a_\alpha:\alpha<\delta\rangle$ in $M$ is indiscernible whe…

Let $T$ be a dependent complete first-order theory, let $M$ be a saturated model of cardinality $\kappa>|T|$, and let $\bold I=\langle a_\alpha:\alpha<\kappa\rangle$ be an indiscer…

Stability conjecture for strongly $2$-dependent groups. Every strongly $2$-dependent group is stable; equivalently, if $G$ is a group such that

Frank Wagner's conjecture. There is some $n$ such that, for every finite set $v\subseteq\omega$, the intersection

Let $\varphi(x;y)$ be a partitioned formula. A formula has uniform definability of types over finite sets (UDTFS) if its types over every finite parameter set can be uniformly defi…