207 problems

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

For a permutation $\pi\in S_n$, let $d_T(\pi,\iota)$ be the minimum number of transpositions whose product sends $\pi$ to the identity permutation $\iota$. Define the transposition…

Let $n>0$, and let a connected group $G$ of finite Morley rank act faithfully and transitively on a definable set $X$ of Morley rank $n$. If the action is generically $(n+2)$-trans…

Let $G$ be a finite primitive permutation group on a set $\Omega$, and let $b$ be its base size. Define the generalised Saxl graph $\Sigma_b(G)$ to have vertex set $\Omega$, with d…

Let $\mathcal{V}_n$ and $\mathcal{D}_n$ denote, respectively, the sets of labelled vertex-transitive graphs and vertex-transitive digraphs on $[n]=\{1,\ldots,n\}$, and let…

Let $\mathrm{Sym}(n,k)$ be the set of permutations of $[n]=\{1,\ldots,n\}$ having exactly $k$ cycles. A family $\mathcal{F}\subseteq\mathrm{Sym}(n,k)$ is intersecting if, for every…

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

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…

Neumann–Praeger conjecture. Is there a function $f:\mathbb{N}\to\mathbb{N}$ such that whenever $U<H\leq A$, where $H$ is a normal subgroup of the finite group $A$ of index $n$ and…

Let $G$ be an almost simple primitive permutation group acting on a set $\Omega$, and suppose that its permutation action is binary. Cherlin's almost simple conjecture. Then … This…

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…

Let $q$ be a prime power, and let the collinearity graph of the polar space $\mathcal{Q}^{-}(5,q)$ be the graph whose vertices are the points of $\mathcal{Q}^{-}(5,q)$, with two ve…

Let $\Gamma$ be a graph as in Hypothesis; in particular, $\Gamma$ is a strongly-regular graph in the setting of the paper. Complete classification conjecture. The graph $\Gamma$ is…

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank. Generic transitivity conjecture. Almost all such groups have maximum degree of generic transitivity at…

Intersection density conjectures. (i) If $|\Omega|$ is a prime power, then $\rho(G)=1$. (ii) If $|\Omega|=2p$, where $p$ is an odd prime, then $1\leq\rho(G)\leq 2$. (iii) If…

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 $\mathcal{S}_n$ be the set of permutations of $[n]=\{1,2,\dots,n\}$, with Hamming distance $d_H$. Define $f(n,s)$ to be the minimum size of a subset of $\mathcal{S}_n$ having c…

Fix $0<\delta<1$. Let $\mathfrak{X}$ be a primitive coherent configuration on $n$ vertices, and suppose every constituent has degree at most $\delta n$. Sublinear-rank motion conje…

Deza–Frankl conjecture. For any $k\in\mathbb{N}$ and any $n$ sufficiently large depending on $k$, if $I\subset S_n$ is $k$-intersecting, then

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…