18 problems

Let $G$ be a finite primitive permutation group of degree $n$ with point stabiliser $H$, and let $b(G,H)$ be its base size. Pyber's conjecture. There exists an absolute constant…

Let $G\leqslant \operatorname{Sym}(\Omega)$ be a primitive almost simple permutation group in a non-standard action, meaning that its point stabiliser is non-standard. Let…

Burness–Giudici conjecture. Every two vertices of $\Sigma(G)$ have a common neighbour.

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite permutation group. A base for $G$ is a sequence of points of $\Omega$ with trivial pointwise stabiliser, and $b(G)$ denotes t…

Let $G$ be a finite almost simple group with socle $G_0$, and let a non-standard subgroup mean a subgroup whose associated primitive action is non-standard in the sense defined in…

Let $G$ be a finite almost simple group. A tuple of core-free subgroups $(H_1,\ldots,H_k)$ is regular if, for some $g_i\in G$, one has … Call such a tuple non-standard when all its…

Let $G\leq \operatorname{Sym}(\Omega)$ be a finite permutation group with base size $b(G)\geq 2$, and let $\mathcal{H}(G)$ be its Saxl hypergraph; in the original graph formulation…

Let $G\leq \operatorname{Sym}(\Omega)$ be a primitive permutation group with base size $b(G)=2$, and let $\Sigma(G)$ be its Saxl graph, whose vertices are the points of $\Omega$ an…

Let $n\geq 2k$, let $G={\rm S}_{n,k}$, and let $H$ be a point stabiliser of $G$. Write $b(G)$ for the base size of the permutation group $G$, let $\mathcal{K}(G,H)$ be the Külshamm…

Let $G$ be a primitive soluble permutation group, and let $b(G)$ denote its base size. A greedy base is a base produced by repeatedly choosing a point in a longest orbit of the cur…

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geqslant 2$. For $\alpha\in\Omega$, call a $G_\alpha$-orbit on $\Omega$ almo…

Let $G\leqslant\operatorname{Sym}(\Omega)$ be primitive with base size $b(G)=2$. For $\alpha\in\Omega$, let $N(\alpha)$ be its neighbourhood in the Saxl graph $\Sigma(G)$, and call…

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser. The base-size conjecture says that the base size of $G$ is at most $5$. This is a proposed…

Let $G$ be a finite group with a core-free maximal subgroup, and let $m(G)$ be the minimal index of a core-free maximal subgroup. Let $R_{\rm max}(G)$, $R_{\rm ns}(G)$ and…

Let $G$ be a primitive non-large-base subgroup of $\mathrm{S}_n=\operatorname{Sym}(\{1,\ldots,n\})$, and let $\mathrm{I}(G)$ denote the maximum length of an irredundant base for…

Babai–Goodman–Pyber conjecture.

Logarithmic irredundant-base conjecture. There exists a constant $C>0$ such that, if $G$ is not a primitive large-base group, then

Let $G$ be a finite primitive permutation group of almost simple type acting on a finite set $S$, and suppose that the action is non-standard, meaning that it is not one of the sta…