27 problems

A structure is monadically $IP$ if every expansion by unary predicates is NIP. Monadic NIP trace-definability conjecture. Monadically NIP structures cannot trace define infinite gr…

Braunfeld's monadic NIP orbit-growth conjecture. For a countable $c9$-categorical structure $c4$, $c4$ is monadically NIP if and only if

An NIP field is an infinite field whose first-order theory does not have the independence property. NIP fields conjecture. Every infinite NIP field is either separably closed, real…

Let $M$ be a model of an NIP theory, let $G$ be a $0$-definable group in $M$, and let $N\succ M$ be an $|M|^{+}$-saturated elementary extension. Let $S_G^{\operatorname{fs}}(N,M)$…

Let $K$ be a field whose pure field theory is NIP$_n$, and let $v$ be a henselian valuation on $K$. Write $(K,v)$ for the resulting valued field. Jahnke's NIP$_n$ henselian expansi…

A valued field is strictly NIP$_n$ when its theory is NIP$_n$ but has IP$_{n-1}$, and a valuation is henselian when it extends uniquely to every algebraic extension. The strictly N…

Let $n\geqslant 2$. A pure field is a field considered only in the language of rings, and a theory is strictly NIP$_n$ when it is NIP$_n$ but has IP$_{n-1}$. The NIP$_n$ Fields Con…

Let $G$ be a definably amenable NIP group. An fsg group is a group with finitely satisfiable generics, and a dfg group is a group with definable f-generics. Pillay–Yao's conjecture…

Let $T$ be the theory of divisible ordered abelian groups, equivalently ordered $\mathbb{Q}$-vector spaces, or more generally the theory of ordered $R$-vector spaces for an ordered…

Let $q\geq 2$ be an integer, $M$ be an $L$-structure, and let $\varphi(x,y)$ be an $L(M)$-formula with dual shatter function $\pi_{\varphi}^*(n)\in o(n^q)$. A family of instances…

Let $T$ be an $NIP$ theory, let $M$ be a structure for $T$, and let $G$ be a group definable over $M$. Let $u*\mathcal M$ denote the Ellis group of $G(M)$, let $H$ be the subgroup…

Henselian-valued-field NIP$_n$ conjecture. For every $n\geqslant 2$, no strictly NIP$_n$ henselian valued fields exist.

Pure-field NIP$_n$ conjecture. For every $n\geqslant 2$, no strictly NIP$_n$ pure fields exist; equivalently, a pure field is NIP$_n$ if and only if it is NIP.

Order-or-valuation conjecture. Every unstable NIP field admits a definable field order or a definable non-trivial valuation.

Simon's conjecture. There are only countably many finitely homogeneous NIP structures up to isointerdefinibility.

Classification conjecture. Any strongly NIP ordered field $(K,<)$ is almost real closed.

Let $(R,<,+)$ be a dense archimedean ordered abelian group, let ${\mathscr{R}}$ be a strongly dependent and noiseless expansion, and let ${\mathscr{N}}$ be a highly saturated eleme…

Let $(R,<,+)$ be a dense archimedean ordered abelian group, let ${\mathscr{R}}$ be an expansion of it, and let ${\mathscr{R}}^{\square}$ be the completion structure used in the pap…

Let $M$ be a finitely homogeneous NIP structure. Structural conjecture. The following hold: 1. The automorphism group $\operatorname{Aut}(M)$ acts oligomorphically on the space of…

Assume that $M$ is an $c9$-categorical NIP structure satisfying the additional rank-theoretic condition of rosiness described in the paper. The conjecture compares such structures…

Let $M$ be a finitely homogeneous structure in a finite relational language and assume that $M$ is NIP. The conjecture concerns classification up to bi-interpretability. Classifica…

NIP field conjecture. Every NIP field is either algebraically closed, real closed, or admits a definable Henselian valuation.

A field is NIP if its theory has the non-independence property. The superrosy NIP field conjecture. Every infinite superrosy field with NIP is either algebraically closed or real c…

Assume that $\mathbb K$ has NIP. Let $\mathbb G$ be a definable group that is definably connected, meaning it has no proper definable clopen subgroup, and definably compact, meanin…

Nilpotent-by-finite conjecture. Every $c9$-categorical group with NIP is nilpotent-by-finite.