9 problems

Optimal-constant conjecture. The constant $\frac12$ in this inequality is not optimal; equivalently, there should be a universal constant greater than $\frac12$ for which the inequ…

Independence non-factorization conjecture. Outside those conditions, in general, independence between $Y_1$ and $Y_2$ cannot be achieved; equivalently, the joint characteristic fun…

Let $G$ be a finite simple graph with no isolated vertices. A dominating set is a set of vertices whose closed neighborhood is all of $V(G)$; let $\gamma(G)$ be the minimum size of…

Fix a random matrix model, such as the Bernoulli $(0,1)$ or $(-1,1)$ model, and consider the statistics studied in the paper after reduction modulo a prime $p$. Cross-prime indepen…

For each $k\in{\mathbb N}$, let $\delta_k$ denote the staircase partition used in the paper, and let $s_{\delta_k}$ be its Schur function under the random specialization over…

Let $G$ be a cubic graph, let $n(G)$ denote its number of vertices, and let $\alpha_e(G)$ denote its exponential independence number. Exponential-independence growth conjecture. Th…

Fink–Jacobson conjecture. For every graph $G$ and integer $k \geq 1$,

Independence characterization conjecture. Then

Let $\mathcal{F}$ be a family of subsets of the positive integers, and let a measure-preserving dynamical system (m.d.s.) be called $\mathcal{F}$-independent when it has the indepe…