15 problems

Profile quasi-polynomial conjecture. If $R$ has bounded signature or finite kernel and $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is a quasi-polynomial.

A class of relational structures is independent when it has the relevant independence property, and it is hereditary when it is closed under induced substructures. A TOWS graph is…

Pouzet's conjecture. Every $\omega$-well structured relational structure $\mathcal{A}$ is a reduct of an $\omega$-well structured, totally ordered structure.

Let $\mu$ be a finite relational signature, and let $\mathscr S_{\mu}$ be the class of all relational structures of signature $\mu$ without any finite monomorphic decomposition. A…

A continuous equation scheme is understood in the sense of MPP16, and each such scheme induces a monad on the relevant category of relational structures. Finitarity conjecture. Mon…

Let $R$ be a relational structure with a finite monomorphic decomposition, and let siblings be counted up to isomorphism. Finite monomorphic decomposition conjecture. Under this as…

Let $R$ be a relational structure, and let $\operatorname{sib}(R)$ denote the number of siblings of $R$, counted up to isomorphism. A structure is cellular in the sense used in the…

Finite basis conjecture. There is a finite subset $\mathfrak A$ of pairwise incomparable structures in $\mathscr S_{\mu}$ such that every member of $\mathscr S_{\mu}$ embeds some m…

Let $\mathcal{M}$ be an age-inexhaustible relational structure, meaning that any two finite substructures of $\mathcal{M}$ have disjoint copies within $\mathcal{M}$. Let the Camero…

Quasi-polynomiality conjecture. If $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is eventually a quasi-polynomial.

Quasi-polynomiality conjecture. If $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is eventually a quasi-polynomial.

Equivalence conjecture. Any two regular relational symplectic groupoids integrating $M$ are equivalent.

Pouzet's conjecture. Under weak hypotheses, the generating series of the profile $\varphi_R(n)$ is a rational function whenever the growth of $\varphi_R(n)$ is subexponential.

Let $R$ be a relational structure with profile $\varphi_R$. Assume that $R$ has bounded signature or finite kernel, and that $\varphi_R(n)$ is bounded by some polynomial in $n$. Ev…

Let $R$ be a relational structure with profile $\varphi_R$, and let the kernel of $R$ be finite. Assume that $\varphi_R(n)$ is bounded by some polynomial in $n$. Eventual quasi-pol…