47 problems

Let $p_c(n)$ be the critical probability for the random subgraph of the $n$-cube, let $V=2^n$, and write … Assume $\epsilon\rightarrow0$ and…

Feder–Subi conjecture. Every 2-colouring of $E(Q_n)$ contains a path between some pair of antipodal vertices which changes colour at most once.

Hypercube saturation conjecture. For every $k\geq 2$,

Balanced-event probability conjecture. There exist $C,d_{0}>0$ such that for all dimensions $d\ge d_{0}$,

Consider a length-regular drawing of the $d$-dimensional hypercube, with edge lengths $\ell_1\leq \ell_2\leq\cdots$. Length-profile conjectures. The following assertions are propos…

Let $Q_n$ be the discrete $n$-dimensional hypercube, and let a $k$-player profile be a tuple of $k$ chosen vertices of $Q_n$. A profile is in equilibrium if no player can strictly…

Long-cycle conjecture. Let $\varepsilon>0$ be a constant, and let

Norine's conjecture. For $n\ge 2$, any antipodal edge-coloring of $Q_n$ contains antipodal vertices $u$ and $\bar u$ such that $u$ and $\bar u$ are joined by a monochromatic path.

Let $Q^n$ be the $n$-dimensional hypercube, and let the profile of a subgraph be the tuple recording the number of its edges in each coordinate direction. A tuple is even if every…

Let $Q^n$ be the $n$-dimensional hypercube, and let an admissible tuple be a tuple that occurs as the profile of a matching in $Q^n$. For $\bm{x}=(x_1,\ldots,x_n)\in\mathbb{N}^n$,…

Let $Q_n$ be the $n$-dimensional hypercube. A geodesic path is a shortest path between its endpoints, and a colour change occurs when consecutive edges receive different colours. M…

Let $Q_n$ be the $n$-dimensional hypercube, with antipodal vertices $v$ and $\bar v$ differing in every coordinate. Leader–Long conjecture. In every red-blue colouring of the edges…

Clifton–Huang conjecture. For $k \ge 2$ and $n$ sufficiently large,

Let $n\in\mathbb{N}$ be even, and let $\mathrm{B}^{n}=[-1,1]^n$. Write $\mathcal{Q}(\mathrm{B}^{n})_n$ for the degree-$n$ truncation of the quadratic module generated by…

Let $d$ and $i$ be positive integers with $1\leq i<d$. Define … Let $i(K_{i,d-i})$ denote the inducibility of the complete bipartite graph $K_{i,d-i}$. E(d,i) density conjecture. E…

Let $H$ be a configuration in $Q_d$, and let $\lambda(H,d)$ denote its $d$-cube density. Classification conjecture above $5/8$. If … then either $H$ is layered and…

Let $W_7,W_8,W_9,W_{10}$, and $W_{12}$ be the configurations in $Q_3$ listed in Table, and let their lower bounds there be the corresponding values of their 3-cube densities. The e…

Signs-model conjecture. Among models in $\mathcal{D}_{\geq p}(Q_3)$, the signs model minimises the probability that $Q_3$ is connected.

For $B\subset\{0,1\}^k$ and an integer $n\geq k$, let $\operatorname{ec}(B)$ be the minimum number of hyperplanes whose union intersects the Boolean cube precisely in $B$. Dimensio…

Perfect-path density conjecture. For all $d\geq 3$,

Perfect-cycle density conjecture. For all $d\geq 4$,

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, let $\mathcal{Q}^n_p$ be its random subgraph obtained by retaining each edge independently with probability $p$, and let…

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, and let $\mathcal{Q}^n_p$ be the random subgraph obtained by retaining each edge independently with probability $p$. For fixed…

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, and let $\mathcal{Q}^n_p$ be the random subgraph obtained by retaining each edge independently with probability $p=p(n)$. The…

Nisan–Szegedy's sensitivity conjecture. There exists an absolute constant $C>0$ such that, for every Boolean function $f$,