15 problems

Strong dualizability conjecture. Every measurable equivalence relation on a standard Borel space is strongly dualizable.

Isaev–McKay–Southwell–Zhukovskii conjecture. If $d_2=\omega(1)$, then there exists a coupling of $G_{\oplus}\sim G(n,d_1)\oplus G(n,d_2-d_1)$ and $G\sim G(n,d_2)$ such that

Gao–Isaev–McKay conjecture. There exists a coupling such that

Kim–Vu conjecture. The range of values for which the corresponding inclusion couplings hold should extend to

Let two random graphs be sampled under a coupling of an ERGM and a corresponding Erdős–Rényi model. Localization conjecture. For certain ERGM specifications, under some coupling wi…

Absolutely continuous coupling conjecture. For every $d\in\mathbb{N}$, there exist distributions $f_1,f_2,f_3,f_4$ on $\mathcal{C}$, absolutely continuous with respect to Lebesgue…

Compatible-uniform-coupling conjecture. For every $d\in\mathbb{N}$, the triple

Let $(\Omega_1, \mathcal{F}_1, \mu)$ and $(\Omega_2, \mathcal{F}_2, \nu)$ be probability spaces. A coupling of $mu$ and $nu$ is a joint distribution having these measures as margin…

Let $X_m\sim \operatorname{Binomial}(m,1/2)$ and let $Y_m$ and $Z$ be the coupled variables appearing in Tusnády's inequality. In the bound … where $C,\widetilde{C}>0$ are universa…

Let $N(n)$ be uniformly distributed on $\{1,\ldots,n\}$, with prime factorization … For each prime $p\le n$, let $Z_p$ be an independent geometric random variable with parameter…

Let ${\boldsymbol d}=\boldsymbol d(n)\in\{0,\ldots,n-1\}^n$ be a near-regular degree sequence, meaning that the difference between its maximum and minimum entries is…

Let $n$ be the number of vertices, and let $0\le d_1\le d_2\le n-1$ be integers such that $d_1n$ and $d_2n$ are even. Write ${\mathcal G}(n,d_i)$ for the uniformly random $d_i$-reg…

Let $W_i=(W_i,S_i,I_i,\mathbf{\Omega}_i)$ be signed graphexes on sigma-finite Borel spaces, let $D_{\lvertW_i\rvert}>0$ everywhere, and suppose that the weak kernel distance satisf…

Let $D$ be a bounded domain. Suppose that reflecting Brownian motions in $D$ admit a shy co-adapted coupling. A rigid-motion symmetry of $D$ is a Euclidean rigid motion preserving…

Let $D$ be a bounded contractible domain in a Euclidean space of arbitrary dimension, and consider reflecting Brownian motions in $D$ together with co-adapted couplings. A coupling…