435 problems

Let $S_n$ be the symmetric group, and let $t$-intersecting mean that any two permutations agree on at least $t$ points. For a family $F\subset S_n$, write $|F|$ for its cardinality…

Let $\widetilde{\mathscr{D}}$ be the input design, let $e$ and $\widetilde{k}$ be the parameters of Construction $\mathscr{D}(\widetilde{\mathscr{D}},e)$, and let…

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 $\mathfrak{A}$ be a countable homogeneous structure in a finite relational signature. A reduct of $\mathfrak{A}$ is a structure obtained by restricting its definable relations,…

Let $H$ be a finite group, and let $\mathcal{M}(H)$ denote the set of maximal subgroups of $H$. Wall's conjecture. The bound … holds for every finite group $H$. The conjecture woul…

A transitive permutation group $L$ is graph-restrictive if there exists a constant $c=c(L)$ such that, whenever $G$ is an arc-transitive group of automorphisms of a graph $\Gamma$…

Let $G$ be a finite group and let an $S$-ring over $G$ be a subring of $\mathbb{Z}G$ that is a free $\mathbb{Z}$-module spanned by a partition of $G$ closed under taking inverses a…

Tucker's Infinite Motion Conjecture. Every connected, locally finite graph with infinite motion admits an asymmetric 2-coloring.

Kantor's conjecture. Every finite flag-transitive generalized quadrangle is classical, isomorphic to $\mathrm{GQ}(3,5)$, or is the generalized quadrangle of order $(15,17)$ arising…

Let $T$ be a finite simple group, and let $\operatorname{Out}(T)$ denote its group of outer automorphisms. Schreier's conjecture. The group $\operatorname{Out}(T)$ is solvable. Thi…

A permutation group is graph-restrictive if there is a constant such that, for every pair $(\Gamma,G)$ consisting of a graph $\Gamma$ and a vertex-transitive group of automorphisms…

Let $\mathcal{D}$ be a $3$-$(v,k,\lambda)$ design, and let $G$ be a block-transitive automorphism group of $\mathcal{D}$. The design is point-imprimitive if the action of $G$ on it…

Let $Q=(Q,\star)$ be a quandle, meaning that $a\star a=a$ for every $a\in Q$ and every left translation $L_a\colon Q\to Q$, $x\mapsto a\star x$, is an automorphism. The left multip…

Let $G$ be a finite simple transitive permutation group of degree $n$, and let $\delta(G)$ denote the proportion of elements of $G$ that are derangements. Boston–Shalev conjecture.…

Let $G$ be a permutation group on a set $\Omega$. For a positive integer $n$ and tuples $I=(\omega_1,\ldots,\omega_n)$ and $J$ in $\Omega^n$, write…

Let $\Pi$ be a finite projective plane admitting a group of automorphisms that acts transitively on its points. Point-transitive projective plane conjecture. The projective plane…

Almost all almost simple groups conjecture. Almost all almost simple groups are minimal wexp-nonsolvable.

Polycirculant conjecture. Every vertex-transitive (di)graph is an $m$-Cayley (di)graph for some positive integer $m$; equivalently,

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 $G$ act primitively on an infinite set $\Omega$ and suppose that $G$ has a finite suborbit whose pair is also finite. The subdegrees of $G$ are then all finite. Exponential-gro…

Cameron's conjecture. The $t$-umvirate families are the only maximum-sized $t$-intersecting families of permutations. The supplied context says that this conjecture was later prove…

Let $[n]=\{1,\ldots,n\}$, and let a family of permutations of $[n]$ be $t$-setwise intersecting if, for any two permutations $\pi$ and $\pi'$ in the family, there exists a $t$-subs…

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…