18 problems

Let an automorphism be hyperbolic if it has a hyperbolic structure, and let property $(*)$ be the property defined earlier in the paper. A polymorphism is prime if it has no nontri…

Quantitative approximate polymorphism conjecture. The dependence between $\epsilon$ and $\delta$ in the Alekseev--Filmus theorem can be improved to polynomial or even linear.

Flexibility-free approximate polymorphism conjecture. The Alekseev--Filmus theorem should hold without the assumption that $P$ is flexible, for any alphabet $\Sigma$: for every suf…

Let $\mathcal{D}$ be a finite relational structure, and let the algebraic condition discussed above be the condition that every polymorphism operation of $\mathcal{D}$ has the stat…

Let $(M,\mu)$ be a space with a continuous $\sigma$-finite measure, and let $\operatorname{Gms}(M)$ be the group of nonsingular transformations of $M$. For spaces $(X,\xi)$ and…

Let $\mathbf{A}$ be the core of a reduct of a finitely bounded homogeneous structure. A pseudo-Siggers identity is an identity of the form … where $s$ is a 6-ary polymorphism and…

A constraint language has polymorphisms, operations preserving all its relations. The relevant identities are equations satisfied by a family of such polymorphisms; a set of identi…

Let $n\geq 4$, let $\mathbb{G}$ be an $n$-cycle, and let $f$ be a surjective homomorphism from a product of subpaths of $\mathbb{G}$ of length $n-2$ to $\mathbb{G}$. An edge $e$ of…

Bodirsky's canonical pseudo-Siggers conjecture. If $\mathbb B$ does not have the SOP, then the existence of a pseudo-Siggers polymorphism of $\mathbb B$ implies the existence of a…

Consider Boolean functions $f_0,\ldots,f_m\colon\{0,1\}^n\to\{0,1\}$ and $g_0,g_1,\ldots,g_n\colon\{0,1\}^m\to\{0,1\}$ satisfying … Here, literals are variables or their negations,…

Let $f$ be a Boolean function, and let $x,y$ be independent uniformly random Boolean inputs. A function is correlated with a character if there is some $S$ such that it agrees with…

Let a finite Boolean Promise CSP be a promise constraint satisfaction problem whose input domain is {0,1\} and whose output domain is finite. A polymorphism is a function preserv…

Let ${\mathfrak B}$ be a countable $\omega$-categorical model-complete core. A polymorphism is an operation preserving all relations of ${\mathfrak B}$, and an endomorphism is a un…

Let ${\mathfrak B}$ be a countable $\omega$-categorical model-complete core, and let $f$ be an $n$-ary polymorphism of ${\mathfrak B}$. Cyclic polymorphism conjecture. If, for ever…

Let $Gamma$ be an $omega$-categorical structure with finite relational signature. A family of totally symmetric polymorphisms consists of polymorphisms of every arity whose value i…

Let $Gamma$ be an $omega$-categorical relational structure. A semi-lattice polymorphism is a commutative, associative and idempotent binary polymorphism. The semi-lattice polymorph…

Let $Gamma$ be a finite-signature relational structure with finite domain $D=D(Gamma)$. A $k$-ary polymorphism $f:D^k\to D$ is cyclic when … for all $x_1,dots,x_k\bin D$. The cycli…

The tractability frontier conjecture. The constraint satisfaction problem $\operatorname{CSP}(Gamma)$ is tractable if and only if there exist a $4$-ary polymorphism $f$ of $Gamma$…