3 problems

Joseph's conjectures. One should have $\ell\in\mathbb{Q}$, $x_i\geq\ell$ for all but finitely many $i$, and $\ell\in S$. Equivalently, the second conjecture says that $S$ is well-o…

Antolín, Martino and Ventura's meta-conjecture. For any “reasonable” sequence $M=(\mu_n)_{n=1}^\infty$ of probability measures on $G$, we have $\operatorname{dc}_M(G)>0$ if and onl…

Let $G_{1}$ and $G_{2}$ be finite groups, and let $Pr^{2n}(G)$ denote the generalized commuting probability of $G$ at level $2n$. Write $Z(G)$ for the center of $G$. The conjecture…