116 problems

Let $Q^d$ be the graph with vertex set $\{0,1\}^d$ in which two vertices are adjacent if they differ in exactly one coordinate, and write $n:=2^d$ for its order. Let $Q^d_p$ be the…

Benjamini–Häggström–Mossel range conjecture. If $f\in_r{\operatorname{Hom}}_{v_0}(Q_n)$, then

Let $H_n=[0,1]^n$ be the hypercube, described by $g_i=x_i-x_i^2\geq 0$ for $i=1,\ldots,n$. For even $n$, let $M_{n,n}(\mathbf g)$ be the degree-$n$ truncated quadratic module gener…

Let $Q^t$ be the binary hypercube and let $Q^t_p$ be the random subgraph obtained by retaining each edge independently with probability $p$. A giant component is a connected compon…

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

Let $Q_n$ be the poset of all subsets of $\\{1,\ldots,n\\}$ ordered by inclusion. A chain decomposition partitions $Q_n$ into chains, and two decompositions are orthogonal if every…

Let $Q^d$ be the $d$-dimensional hypercube, let $V_p$ denote the vertices retained in the site-percolation model, and let $x$ be the constant such that the unique giant component o…

Kahn's sharp range conjecture. The absolute constant in the bound $R(f)=O(1)$ with high probability can be taken to be $5$; equivalently,

Let $Q_d$ be the $d$-dimensional hypercube. A 1-factorisation is semi-perfect if one 1-factor has the property that its union with every other 1-factor is a Hamilton cycle. Craft's…

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

Let $Q_n$ be the $n$-dimensional hypercube with a red-blue edge-colouring, and call a path between antipodal vertices to have a colour change whenever consecutive edges have differ…

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…

Talagrand's conjecture. For any $s>0$, there exists a constant $c_s$ independent of the dimension $n$ such that, for $t>1$,

Let $r\geq 3$ be fixed, and let $Q_d$ be the graph with vertex set $\{0,1\}^d$ in which two vertices are adjacent when they differ in exactly one coordinate. For the $r$-neighbour…

Erdős's conjecture. Any subgraph of $Q_n$ asymptotically containing at most half the edges of $Q_n$ is 4-cycle free.

Let $Q_n$ be the $n$-dimensional hypercube, and let $p_c(Q_n,n/2)$ denote the critical probability for majority bootstrap percolation with threshold $n/2$. Sharp asymptotic conject…

Let ${\mathbb Q}_n$ denote the $n$-dimensional hypercube, and let $p_c({\mathbb Q}_n)$ be its percolation critical value. A scaling window is the range of values of $p$ over which…

Let $p_c(n)$ be the critical probability for the random subgraph of the $n$-cube, let $V=2^n$, and let ${\mathcal C}_2$ denote the second largest cluster. Suppose … with…

Let $p_c(n)$ be the critical probability for the random subgraph of the $n$-cube, let $V=2^n$, and suppose … with $\epsilon\rightarrow0$ and $\epsilon V^{1/3}\rightarrow\infty$. Le…

Let $p\in[0,1]$, define $\epsilon=\epsilon(p)$ by … and suppose $p$ is below the scaling window, so $\epsilon V^{1/3}\rightarrow-\infty$, where $V=2^n$. Let…

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

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

Let $b\in\mathbb{N}_2$ and $\rho\in(1,e)$, and suppose that, for some sufficiently large constant $L_1>0$, uniformly for $m\in[1,d/L_1]$ as $d\to\infty$, the first passage time sat…

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 $Q_d$ be the $d$-dimensional hypercube, let $\mathcal{R}_d$ be the recursively defined drawing described above, and let $CR_{\max}(Q_d)$ denote the maximum rectilinear crossing…