8 problems

Let $G$ be an oligomorphic permutation group whose profile is bounded by a polynomial. Writing $varphi_G(n)$ for the number of $G$-orbits on $n$-element subsets, Cameron's conjectu…

Let $G$ be an oligomorphic permutation group acting on a domain $\Gamma$, and let $\mathcal{A}^G$ be Cameron's graded $\mathbb{Q}$-algebra of $G$-invariants in the incidence algebr…

Ramsey-ordering conjecture. There exists such a Ramsey ordering for which, for all $\beta,\beta'\in\mathcal{B}$ and $\gamma,\gamma'\in\mathcal{C}$,

Bodirsky–Schneider–Thom conjecture. If $G\curvearrowright\Omega$ is oligomorphic, then all mixed identities of $G$ are singular.

Singularity conjecture. Every mixed identity for the group $\operatorname{Aut}(X,R)$ is singular.

Let $G$ be a primitive oligomorphic permutation group acting on a countable infinite set … of size $n$. Macpherson's theorem states that there is a constant $c>1$ such that either…

Let a structure be aleph0-categorical if it has exactly one countable model up to isomorphism, and stable in the model-theoretic sense. A structure is aleph0-stable if, over every…

Let $G$ be a permutation group with no finite orbits, and let $\mathcal{A}^{G}$ be its Cameron algebra. Suppose in addition that $G$ is oligomorphic, meaning that it has finitely m…