14 problems

Characterization conjecture. A family of dependencies $\mathcal D$ is strongly first order if and only if every dependency $\mathbf D\in\mathcal D$ is strongly first order.

Łoś's conjecture. $Cat(T)$ contains every uncountable cardinal or no uncountable cardinal.

For a graph class $\mathcal{C}$, say that it has bounded merge-width when its merge-width is bounded by a constant. A first-order transduction is an interpretation of graphs from s…

Let a first-order sentence $\phi$ define the hereditary first-order model-checking problem $\operatorname{Her}(\phi)$. A problem is coNP-intermediate if it belongs to coNP but is n…

Let $Th'$ be a theory in the empty-signature first-order language, $Th' subseteqmathcal{FO}^emptyset$, such that … Here, $\mathcal{S}_c$ is the relevant Nelsonian semantic structur…

Let $0\leq d<\frac{1}{2}$, and let $\psi$ be a first-order sentence in the language of groups. Let $F_k$ be the free group of rank $k\geq 2$, and consider the density-$d$ random-gr…

Sublogarithmic distinguishability conjecture. With high probability,

Completeness conjecture. An axiomatization for the one-variable fragment of the first-order version of full Lambek Calculus should be complete for valid equations, but not for cons…

Let $T_{1}$ be the base theory for list concatenation, and let $\mathrm{Open}(\mathcal{L}_{1})$ denote the set of open formulas in its language. Let…

Malliaris–Shelah's conjecture. The maximal class is exactly the class of SOP$_2$ first-order theories.

In the calculus, let $\mathrm{D.FO}$ denote the present first-order deductive framework, and let the prelinearity rule be … The two rules in $(\ast)$ are the analytic structural ru…

Disjunctive-binding decidability conjecture. The fragment $db$ enjoys a decidable satisfiability problem.

Conjunctive-binding positive-properties conjecture. The fragment $cb$ enjoys the finite-model property and has a decidable satisfiability problem.

Extension-preservation conjecture. Some of these extensions preserve the same model-theoretic and algorithmic properties.