16 problems

Street–Whitehead's conjecture. The finite dihedral group of order $2n$ is not filled when

Let $A_n$ be an alternating group, and call a subset $X\subseteq A_n$ product-free if there are no $x,y,z\in X$ with $xy=z$. A maximal subgroup is a proper subgroup maximal under i…

Let $G$ be a compact group equipped with its Haar probability measure, and let a Kedlaya set be a set of the form … where $G$ acts on a set $X$, $B\subseteq X$, and $x\in X$. A mea…

For $\mathrm{SU}(n)$, let the maximal Haar measure of a measurable product-free subset be the supremum of the Haar measures of measurable subsets $A\subseteq \mathrm{SU}(n)$ satisf…

Let $G$ be a finite group, and call a subset $A\subseteq G$ product-free if $gh\notin A$ whenever $g,h\in A$. Babai–Sós conjecture. There exists an absolute constant $c>0$ such tha…

Extremal product-free set conjecture for free groups. It is possible to choose such a labelling so that $S\subset T$.

Stability conjecture. For each $\delta>0$, there exists $\varepsilon>0$ such that, whenever $S\subset\mathcal{F}$ is product-free and

Extremal 3-product-free set conjecture. One of the following holds: either the letters of $\mathcal{A}$ can be labelled by $\mathbb{Z}/3\mathbb{Z}$ so that $S$ is contained in the…

Extremal $5$-, $7$-, and $13$-product-free set conjecture. For $k=5,7,13$, the extremal sets are exactly the sets described by the paper's structural theorem for $k$-product-free s…

Leader–Letzter–Narayanan–Walters conjecture. If $S\subset\mathcal{F}$ is product-free and $d^{\ast}(S)=1/2$, then

Let $A_n$ be the alternating group, and call a subset $F\subseteq A_n$ product-free if there are no $a,b,c\in F$ with $a=bc$. For $x\in\{1,\ldots,n\}$ and…

Upper Banach density conjecture. No product-free subset $S\subset\mathbf{F}_{\mathscr{A}}$ has upper Banach density exceeding

Uniqueness conjecture. If $S\subset\mathcal{F}_{\mathscr{A}}$ is product-free and

Classification conjecture. The group $G$ is filled if and only if $G$ is either an elementary abelian $2$-group or one of

Let $G$ be a finite group, let $\mathbb{F}$ be a field, and let $X_1,\dots,X_k$ be subspaces of the group algebra $\mathbb{F}[G]$ satisfying … Define $M_k(G)$ to be the minimum, ov…

Let $G$ be a group, and let $S\subseteq G$ be a locally maximal product-free set of size $3$, meaning that $S$ is product-free and is not properly contained in any larger product-f…