111 problems

Let $C_n$ be the cycle with $n$ edges, and let $q(C_n)$ denote its consistency probability. Define … The cycle consistency probability conjecture. The consistency probability of…

For $0\leq\mu_1\leq\cdots\leq\mu_n$ and $\lambda>\sum_{i=1}^n\mu_i$, let $X_1,\ldots,X_n$ range over independent nonnegative random variables satisfying $\mathbb{E}X_i=\mu_i$. Defi…

Optimal Power of Few conjecture. There is a function $f(x)$ such that $f(x)>1/2$ for $x>0$, $f(x)$ tends to $1$ as $x$ tends to infinity, and, in majority dynamics on $G(n,p)$ with…

Let $D70E_n$ be a uniformly random permutation in the symmetric group $D70E_n$, and let $D43E\operatorname{Pop}$ denote one step of Pop-Stack Sorting, which reverses every maximal…

Simultaneous multipartite subhypergraph conjecture. There exists a partition of $V$ into $r$ classes $V_1,\dots,V_r$ such that, for every $i=1,\dots,\ell$, at least

Let $RH(r,s,A)$ be a Robin Hood game with resource functions $r,s$ and index set $A$. A random strategy is independent of the index set $A$ if its choices do not depend on $A$. Old…

Independent-set density existence conjecture. For every $c>0$ and $r\geq 3$, the limits

MAX-SAT limiting function conjecture. For every $k$ and every constant $c>0$, as $n\rightarrow\infty$,

Satisfiability threshold conjecture. For each $k$ there exists a threshold density $c_k$, such that for any positive $\varepsilon$, for all $c<c_k-\varepsilon$, $F(n,cn)$ is almost…

Let $\mathcal{G}(n,k)$ be the probability space of $k$-majority tournaments on vertex set $[n]$ obtained by uniformly choosing, with replacement, $2k-1$ linear orders of $[n]$. Let…

Let $\Sigma$ be a finite alphabet, let $k\in\mathbb{N}$, and let $\mu$ be a pairwise-connected distribution over $\Sigma^k$ in which every atom has mass at least $\alpha$. A functi…

Let $p=\omega(1/\sqrt n)$ with $p\leq 1/2$, and let $G$ be sampled from either $\mathbb{G}(n,p)$ or $\mathbb{G}(n,1-p)$. A graph is $k$-vertex-minor universal if every graph on any…

General mandatory and blocking edge expectation conjecture. The expected proportions of edges belonging to every maximum-size matching and of edges belonging to no maximum-size mat…

Mandatory and blocking edge expectation conjecture. The expected proportions of edges belonging to every maximum-size matching and of edges belonging to no maximum-size matching sa…

Expected-number conjecture. The expected number of letters drawn is given by these two equivalent expressions.

The generating-function conjecture. The generating function $f(z)$ is given by the displayed expression above.

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\{y:y\not\sim x\}$ and define the range parameter … For distinct $x,y\in P$, let…

Let $P$ be a finite poset, let $w(P)$ denote its width, and for distinct elements $x,y\in P$ define … with … where the probabilities are over a uniformly random linear extension of…

Let $P$ be a finite poset. For distinct elements $x,y\in P$, define … and let … where the probabilities are taken over a uniformly random linear extension of $P$. The 1/3-2/3 conje…

Let $P$ be a finite poset, let $F$ be uniformly distributed in its order polytope, let $H(x)=\mathbb E F(x)$, and let $\operatorname{Win}(x)=\mathbb E[R(x)-Q(x)]$, where…

Let $P$ be a finite poset with $n$ elements, let $x,y\in P$ with $y\ne x$, and let $h_P(y)$ denote the expected position of $y$ in a uniformly random linear extension of $P$. The l…

Let $P$ be a finite poset. For $x\in P$, let $\Pi(x)=\\{y\in P\setminus\\{x\\}:y\not\sim x\\}$, let $\pi(x)=|\Pi(x)|$, and let $\pi(P)=\max_{x\in P}\pi(x)$. Let $\delta(P)$ denote…

Let $P$ be a finite poset, let $w(P)$ denote its width, and let $\delta(P)$ be the maximum, over distinct $x,y\in P$, of $\min\\{p(x\prec y),p(y\prec x)\\}$. The Kahn–Saks conjectu…

Let $P$ be a finite poset that is not a chain. For distinct $x,y\in P$, write $\delta_{xy}=\min\\{p(x\prec y),p(y\prec x)\\}$ and $\delta(P)=\max_{x\ne y}\delta_{xy}$. The 1/3–2/3…

Let $G$ be a group of size $n$, and let $G(p)$ be its random Cayley graph, formed by selecting each equivalence class $\{x,-x\}$ independently with probability $p$. Write…