9 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 $(K,v)$ 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 $R$ 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…

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…