16 problems

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 $K^{(k)}_\ell$ 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…

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…

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

Quadratic edge-count conjecture.

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…

Let $X^{(\ell)}_k$ denote the number of lineages in the ancestry of a leaf after moving $k$ steps away from that leaf in a uniformly sampled ranked tree-child network with $\ell$ l…

Let $P_n$ be the path graph on $n$ vertices. For a weight vector $x$ on $P_n$, define … where $\overline{x}$ denotes the average-weight vector. Path-graph averaging conjecture. For…

Let $S_n$ be the star graph on $n+1$ vertices, and let $x$ be the weight vector on $S_n$ that gives weight $1-\frac{1}{n+1}$ to the central vertex and weight $-\frac{1}{n+1}$ to ev…

Let $(X_{i,j})_{i,j\ge 1}$ be an array of independent, identically distributed exponential random variables of rate $1$. For each $n\ge 2$, let $\bm{V}_n$ denote the vector of last…

Let $\gamma$ be an unbounded one-sided curve in the Euclidean plane, and let $\Omega$ be a simply connected, open, bounded planar domain. Reroot $\gamma$ at a root chosen in…

Fix a monotonically increasing sequence $\{a_n\}$. In the additive decomposition process, a stick whose length belongs to $\{a_n\}$ does not decompose further; sticks of length $1$…

Consider an additive decomposition process starting from a stick of integer length $L$: choose uniformly an integer cut point $c\in[1,L-1]$, replace the stick by pieces of lengths…

Consider the toy genetics process in which an amoeba has a DNA configuration of length $n$, and each generation produces two offspring through independently activated crossover sym…