9 problems

Propagation-time interval conjecture. For every $d\geq 2$,

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. Let the collapsibility number of a…

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. For $r\in\{2,3\}$, write…

A symmetric chain decomposition (SCD) of the hypercube $Q_n$ is a partition of its vertices into symmetric chains, where a symmetric chain is a path $(x_k,x_{k+1},\ldots,x_{n-k})$…

Let $G_{k,2}$ be the bipartite graph with vertex classes $\binom{[n]}{k}$ and $\binom{[n]}{2}$, where a $k$-element set $S$ is adjacent to a 2-element set $T$ exactly when…

Let $Q_n$ be the $n$-dimensional hypercube, let $\mathcal{S}=(a_1,\dots,a_n)$ be a signature of $Q_n$, and let ${\mathcal{E}(\mathcal{S})}$ be the edge slide graph induced by the s…

Let $Q_n$ be the $n$-dimensional hypercube, and let $Q_{n,[a,b]}$ denote the subgraph induced by the vertices whose levels lie in $[a,b]$. For integers $k\geq 1$ and…

Daniel–Jolliffe conjecture. For every $i$, the multiplicity of $2^i$ as a $2$-elementary divisor of $A$ equals the number of eigenvalues of $A$ with exact $2$-power divisor…

Let $n$ be an even positive integer, and let $k$ be a positive integer. The $n$-dimensional hypercube $\mathcal{Q}_n$ has vertex set $\{0,1\}^n$ and edges joining vertices at…