13 problems

Dixon's conjecture. As $|G|$ tends to infinity, the probability that two randomly chosen elements generate $G$ tends to $1$:

Let $T_n$ be the full transformation monoid on an $n$-element set. A monoid is synchronising if it contains a constant mapping. Cameron's conjecture. The probability that two rando…

Consider the Markov chain on the linear extensions of an $n$-element poset that chooses uniformly $i<j$ and successively attempts adjacent switches for $k=i,\ldots,j-1$. Ayyer–Schi…

Let $\mathsf B_{1000}$ be the Boolean lattice on $1000$ generators, and call a 50-dimensional vector $(h_1,\dots,h_{50})$ a generating vector if its components generate…

Let $G$ be a finite group. Let $\mathcal{V}(G)$ be the least positive integer $k$ such that $G$ is generated by $k$ independent uniformly random elements with probability at least…

Dixon's two-generator conjecture. As $\lvert G\rvert\to\infty$, one has $k(Z)=2$ with high probability. This conjecture was verified by Liebeck and Shalev, so the assertion is solv…

Primitive-subgroup conjecture. Almost surely, $\langle\pi,\pi'\rangle$ is not contained in any primitive subgroup apart from $S_n$, $A_n$, and any conjugate of…

Let a degree sequence be power-law distribution-bounded with exponent $\gamma$ when it belongs to the power-law distribution-bounded class described in the source. Gao–Wormald's co…

Let a switch Markov chain be the chain on realizations of a degree sequence that performs a switch, for an unconstrained, bipartite, or directed degree sequence. A degree sequence…

Let $\Gamma$ be a Fuchsian group, namely a finitely generated non-elementary discrete group of isometries of the hyperbolic plane. Let $G$ be a finite simple classical group of suf…

Let $S_k$) be the symmetric group on $k$ elements, and choose two elements independently and uniformly at random from $S_k$. Netto's conjecture. As $k$ tends to infinity, the proba…

Let $n$ be the size of a random distance-hereditary split tree, and let the number of its clique nodes and star nodes be measured under the cycle-pointed sampling framework. Linear…

Lattice-generation conjecture. For every $n\in\mathbb{N}$, there exists a constant $0<c_n<1$ and a rational function $f_n\in\mathbb{R}(x,y)$ satisfying