25 problems

Kahn's conjecture. If $|X|=|Y|=|Z|$, then

General CNF–DNF correlation conjecture. Under these assumptions, the inequality referred to in the source as the CNF–DNF inequality holds.

Let $p_3(n)$ denote the third critical probability at which the relative covariance of paths in $\vec G(n,p)$ changes sign, where $p_1(n)<p_2(n)<p_3(n)$ are the three critical prob…

Friedgut–Kahn–Kalai–Keller antipodal correlation conjecture. For every increasing $f,g:\{0,1\}^n\to\{0,1\}$, if $g$ is antipodal, then

Friedgut–Kahn–Kalai–Keller spectral correlation conjecture. For any increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$,

Kalai–Keller–Mossel correlation conjecture. For every increasing and balanced family $\mathcal{A}\subseteq\Omega_n$, there exists an increasing linear-threshold family…

Let $G=(V,E)$ be a graph and let $u,v,w\in V$. Let $\mu^{\text{F}}_{G,\lambda}$ denote the arboreal gas measure with parameter $\lambda$, and write…

Let $\mu$ be an ERGM measure on graphs with vertex set $[n]$, and let $e_1,e_2,e_3$ be edges spanning a triangle in $K_n$. Let $e$ be an arbitrary edge. Triangle moment conjecture.…

Extremal intersection conjecture.

Monotonicity conjecture. For $\beta_1,\beta_2\in(0,1)$ with $\beta_1<\beta_2$, there exists $\varepsilon>0$ such that

Let $n\in\mathbb{N}$ and let $p_1,\dots,p_n\in 2\mathbb{N}$. For a real centered Gaussian vector $(X_1,\dots,X_n)$, the statement $\mathbf{GPI}_n(p_1,\dots,p_n)$ means … with equal…

Sahi's coefficient-positivity conjecture. For every $n\geq 1$, the coefficient $c_n$ satisfies

Let $L$ be a distributive lattice equipped with a probability measure $mu$ satisfying … for all $a,b\in L$; such an $L$ is an FKG poset. Let $f_1,\ldots,f_n$ be positive monotone d…

Let $M_n$ be the finite honeycomb-lattice region. For $k=1,2,3$, choose sets $A_k,B_k\subset M_n$, and let $U_k$ be the event that fluid $k$ percolates between $A_k$ and $B_k$. Gen…

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the…

Let $M_n$ be the finite honeycomb-lattice region with four-colored cells, and let $B_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates between opposite sid…

Let $\gamma_n$ be the Gaussian probability measure on $\mathbb{R}^n$, and let $K_1$ and $K_2$ be symmetric convex bodies in $\mathbb{R}^n$. The Gaussian Correlation Conjecture. The…

Let $\mathbb{S}^n$ be the canonical Riemannian sphere, let $o$ be the center of a fixed hemisphere, and write the hemisphere as $B(o,\pi/2)\subset\mathbb{S}^n$. Let $K_1,K_2$ be ge…

Let $G$ be the annealed randomly oriented Erdős–Rényi graph $\vec G(n,p)$, with distinct vertices $a,s,b,t$, and let $\{s\to a\}$ and $\{t\to b\}$ be the events that directed paths…

Let $X$ be a finite set, let $\mu$ be an FKG probability measure on $2^X$, and let $p(A)=p_1(A)t+p_2(A)t^2+\cdots$ be a formal series whose coefficients are nonnegative monotone fu…

Single-sign-change conjecture. For any fixed $n\ge 5$, the covariance in $G(n,m)$ changes sign only once between two values of $m$.

Half-probability positivity conjecture. For $p=\frac12$ and $n\ge 6$, the relative covariance $R_n(p)$ is positive.

The three-sign-change conjecture. For $n\ge 27$, the relative covariance $R_n(p)$ changes sign at three critical probabilities $p_1(n)<p_2(n)<p_3(n)$.

Degree-two vertex conjecture. If $s$ has degree at most two, then the events $\{a\to s\}$ and $\{s\to b\}$ are independent or negatively correlated.

Let CNA and CNA+ denote the corresponding conditional negative-association properties, and let h-NLC and h-NLC+ denote the hereditary negative-lattice-condition properties; Rayleig…