202 problems

Uniform boundedness conjecture. There exists an absolute constant $c$ such that $\rho_0(G)<c$ for all finite primitive permutation groups $G$. This conjecture asserts that the larg…

Let $G$ be a transitive permutation group of degree $n$. For a transitive permutation group $K$ of degree $n$, let $I(n)$ be the maximum intersection density among such groups. If…

Let $G=\operatorname{Sym}_n$ and let $T$ be the set of transpositions, so that $\operatorname{Sym}_n(T)$ is the corresponding Cayley graph, $B_r(\sigma)$ denotes the radius-$r$ bal…

Let $G$ be the graph whose vertices are the elements of the symmetric group $S_n$, with two vertices $\pi$ and $\pi'$ adjacent when $\pi(i)\neq\pi'(i)$ for every $1\leq i\leq n$. A…

Let $G$ be an infinite primitive permutation group whose subdegrees are all finite, and let $(m_r)$ be its lower subdegree sequence. Permutation-ends conjecture. If $(m_r)$ grows e…

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

Cameron's integral-domain conjecture. If $G$ has no finite orbits, then $A(G)$ is an integral domain.

Cameron's prime-element conjecture. If $G$ has no finite orbits, then $\varepsilon$ is a prime element in $A(G)$.

Let $(a_0,a_1,\dots,a_l)$ be a sequence generated by the CCL procedure, and let $\pi$ be the fast forward permutation coded by this sequence. Choose $x\in\{0,\dots,N-1\}$ uniformly…

Let $p$ be a prime greater than $7$, let $G=PGL_2(p)$, and let $\Omega$ be the permutation domain of the action under consideration. The relational-complexity conjecture asserts th…

The construction associates generators $\rho_1,\rho_2,\rho_3$ to a group $G$ by … A triality construction conjecture. Apart from two cases, up to coloring symmetries, the associate…

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

Ellis–Friedgut–Pilpel conjecture. For any $n\in\mathbb N$ and $1\leq t\leq n$, the maximum size of a $t$-intersecting family in $S_n$ is attained by one of the families…

Reflection-permutation conjecture. The reflection permutation $\pi$ gives the true minimum for all $(q,n)$, namely

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…

Rank-two conjecture. If $m\geqslant n\geqslant 2$ and

Let $G\leq S_n$ be a transitive permutation group. The diameter of $G$ is defined using the maximum over generating sets, namely…

Let $n>1$. For each $m\in\{2,3,\ldots,n\}$, let $CT_m$ be the group generated by all horizontal class transpositions $\tau_{r_1(m),r_2(m)}$, and define … Let…

Let $n\geq 3$ and let $\vec{p}$ be a regular parameter vector. A label is neutral if $p_{c,i}=p_{i,c}=1/2$ for every $i\neq c$. Let … be the number of neutral labels, and let…

Fill's characterization conjecture. The following are equivalent:

Let $S_N$ be the classical permutation group and $S_N^+$ Wang's quantum permutation group, with $S_N\leq S_N^+$ denoting the inclusion as a quantum subgroup. The Maximality Conject…

Let $S$ be a simple group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $S$. A group element is a $p$-element if its order is a power of $p$. Conder's conjecture. If…

Let $G$ be a finite primitive non-regular permutation group of degree $n$. Write $\operatorname{fix}(g)$ for the number of points fixed by $g\in G$. Neumann's conjecture. There exi…

Let $G$ be a finite group with subgroups $U$ and $U'$, and let $n=|G:U'|$. For a subgroup $V$ of $G$, write $V^g=g^{-1}Vg$ and let $\bigcup_{g\in G}V^g$ denote the union of its con…