52 problems

Let $K^{(k)}_\ell$ be the $k$-uniform complete hypergraph on $\ell$ vertices, and let $R_n(K^{(k)}_\ell)$ denote the final number of edges in its removal process. Bennett–Bohman fo…

Logarithmic path-length conjecture. The expected maximum directed path length of the resulting task-dependency graph is

Quadratic edge-count conjecture.

Let $\mathbf{d}_n$ be a graphic degree sequence, and let $G^P_{\mathbf{d}_n}$ be the final graph of the $\mathbf{d}_n$-process conditioned on having degree sequence $\mathbf{d}_n$.…

Sine trajectories. As the number of particles tends to infinity, every scaled particle trajectory should be uniformly approximated by a sine curve with random amplitude and phase.…

In the random Bulgarian solitaire model with total size $N$, let the typical deviation refer to the deviation of a configuration from the limiting triangular shape. Typical-deviati…

Wormald's conjecture. The random $d$-regular graph and the random $d$-process cannot be distinguished with high confidence.

Let $Z$ be a $d$-dimensional log-concave random vector. The Sudakov minoration principle gives a lower bound for the expected supremum of a process indexed by a finite set of point…

Condition on the event that the cluster of the origin is infinite. Write the edges in the cluster as $\{\ldots,e_{-1},e_0,e_1,\ldots\}$, define…

For the random payment-failure process on a complete graph with all edges having the same capacity, let the currently known upper and lower bounds on the time until the first payme…

Consider a fixed finite number of agents whose values increase in unit steps, with the waiting times between successive increases given by independent identically distributed rando…

Let $\Sigma$ be the simplex of limiting edge-traversal proportions, let $V$ be its set of vertices, and let $\mathcal{D}_0=\{\Theta\in\Sigma\setminus\partial\Sigma\}$,…

Let $M_n$ be the length of the largest of the $n+1$ subintervals obtained after $n$ steps of Kakutani's random interval-splitting process, and let $\sigma^2=16\log 2-10$. Theorem g…

Throughout the paper, the initial opinions are i.i.d. random variables, with expectation $\mu:=\mathbb{E}[f_0(v)]$. Let $f_t(v)$ denote the opinion at vertex $v$ at time $t$, and s…

Let $k\geq2$, and let $\mathcal F$ be a strictly $k$-balanced $k$-uniform hypergraph with $k$-density $\rho$. Write $v(\mathcal F)$ and $e(\mathcal F)$ for its numbers of vertices…

Let $F_n(\mathcal F)$ denote the final number of edges in the $\mathcal F$-free process on $n$ vertices. For graphs $\mathcal F$, suppose the known high-probability lower bound for…

Let $p=\frac{c\log(n)}{n}$, and let $G$ be a random simple temporal graph with edge probability $p$, where $c\in(0,1)$. The largest temporal clique is bounded above by…

Fix $a\in\mathbb R$, and let … Let $R_n(a)$ denote the number $\rho_n(b_n)$ of records associated with the threshold $b_n$. Limiting-law conjecture. As $n\to\infty$, $R_n(a)$ conve…

Contiguity conjecture. The final graph of the random $d$-process is contiguous to the uniformly random $d$-regular graph.

Consider the vertex-reinforced jump process (VRJP) on regular trees with wired boundary conditions. An intermediate phase would be a phase between the regimes identified for the mo…

Let $\vec{z}_t$ denote the disagreement component of the opinion vector at time $t$, and let $\lambda$ be the parameter appearing in the model. Here $\mathbb{E}$ denotes expectatio…

Edge-addition success-probability conjecture.

Let $D_1\ge\dotsb\ge D_n$ be a uniformly random graphic sequence of length $n$. The scaling-limit conjecture. There exists a random continuous function $D:[0,1]\to\mathbb{R}$ such…

Let $D$ be a deck with $n$ cards of each type, and let $\lambda_D$ denote the expected number of steps to absorption for the random card-shuffling process started from $D$. As…

Largest-gap order conjecture. The correct order for $M{ un}$ is