13 problems

Let $T$ be a countable theory and let $f_T(kappa,lambda)$ be the associated two-cardinal counting function for infinite cardinals $kappaleqlambda$. Adler's counting-spectrum conjec…

Let $varphi(x,y)$ be a partitioned formula, and let $f_{varphi}(k,l)$ denote the number of pairwise-inconsistent partial $varphi$-types of size $k$ over a parameter set of size $l$…

Let $varphi(x,y)$ be a partitioned formula. A formula is NTP$_2$ when it does not have the tree property of the second kind, and it is resilient in the sense used in the paper. ULC…

Let $varphi(x,y)$ be a partitioned formula, and let $f_{varphi}(k,l)$ count pairwise-inconsistent partial $varphi$-types of size $k$ over a parameter set of size $l$. Write…

A theory $T$ is called NTP$_2$ if it has no formula with the tree property of the second kind. The Uniform Local Character over Finite Sets conjecture. Every NTP$_2$ theory satisfi…

Let $(K,v)$ be a henselian valued field of positive characteristic. A valued field is tame if it satisfies the standard tameness conditions for henselian valued fields. NTP2 transf…

Let $T$ be an $\text{NTP}_2$ theory, and let burden be the model-theoretic rank on partial types. A rank is sub-additive if the burden of a concatenated tuple is at most the sum of…

Let $R$ be a $W_n$-ring on an algebraically closed field. Wn-ring conjecture. The ring $R$ is NTP$_2$, and the burden of $R$ is at most $n$. The source gives no further context or…

Let $T$ be a first-order theory. Recall that $T$ is resilient if, whenever $(a_i)_{i\in\mathbb{Z}}$ is an indiscernible sequence and $\varphi(x,a_0)$ divides over $a_{\neq 0}$, the…

Definable-envelope conjecture. If $H$ is nilpotent (respectively, soluble), then there is a definable nilpotent (respectively, soluble) group finitely many of whose translates cove…

Valued-field NTP2 conjecture. Then $(K,v)$ is $\operatorname{NTP}_{2}$ (respectively, strong) if and only if $k$ is $\operatorname{NTP}_{2}$ (respectively, strong).

PRC and PpC NTP2 conjecture. A PRC field is $\operatorname{NTP}_{2}$ if and only if it is bounded. Similarly, a PpC field is $\operatorname{NTP}_{2}$ if and only if it is bounded.

In a first-order theory $T$, let $\operatorname{bdn}(a)$ denote the burden of a tuple $a$, and say that $T$ is NTP$_2$ when it has no inp-pattern of depth $\omega$ in a single vari…