7 problems

Let $\mathcal{K}=\Braket{K,+,\cdot,<,\dots}$ be a definably complete expansion of a real closed field. The pigeonhole principle conjecture. $\mathcal{K}$ has the pigeonhole princip…

NIP real field classification conjecture. Every NIP real field is real closed, or admits a non-trivial definable henselian valuation.

Finite-tower proof-length conjecture. There is a finite tower of exponentiations that gives an upper bound on the lengths of proofs of true sentences in $\mathsf{RCF}$.

Let $R$ be a real closed field. For a smooth algebraic variety $X$ over $R$, write $X^{\operatorname{h}}$ for its associated space of real points. A generalized connected open subs…

Let $\mathcal R$ be a d-minimal expansion of $\bar{\mathbb{R}}$. Let $X \subseteq \mathbb{R}^n$ be a manifold definable in $\mathcal R$, and let $Y \subseteq X$ be clopen in $X$. C…

A Turing degree is low$_n$ if its $n$th Turing jump has the same degree as the $n$th jump of the computable degree. Consider spectra of structures, namely the sets of Turing degree…

Let $\phi_{IP}$ denote the choice principle asserting the existence of integer parts for real closed fields. Let $AC$ denote the axiom of choice. Non-implication conjecture. The pr…