5 problems

DMS's conjecture. There exists an interval $X$ of positive integers, not containing $1$, such that no primitive group $G$ acting on any domain $\Omega$ satisfies

Let $\mathcal{I}(G,\Omega)$ denote the set of cardinalities of irredundant bases of a permutation group $G$ on $\Omega$. Forbidden-interval conjecture. There exists an interval $X$…

Let $G$ be a primitive permutation group on a domain $\Omega$, and let $\mathcal{M}(G,\Omega)$ denote the set of cardinalities of minimal bases of $G$ on $\Omega$. Primitive interv…

Let $X\subseteq\mathbb{N}$ with $1\notin X$. For a permutation group $G$ acting transitively on a domain $\Omega$, let $\mathcal{M}(G,\Omega)$ denote the set of cardinalities of mi…

Let $n\geq1$. For each ordered set partition $\Pi$ of $[n]$, let $\mathcal{B}_\Pi$ be the set of monomials constructed from the associated permutation, Grassmann monomial, and sche…