67 problems

Sharp sumset lower-bound conjecture. The inequality

Let $\vv{Q_d}$ denote the oriented $d$-dimensional hypercube, and let $\vv{r}(\vv{Q_d})$ be its oriented Ramsey number. Directed Burr–Erdős conjecture. There is an absolute constan…

Let $Q_r$ be the $r$-dimensional hypercube, let $\alpha_{\text{bal}}(Q_r)$ denote the maximum total size of a balanced bipartite biindependent pair in $Q_r$, and let $a(r)$ be the…

Let $Q_d$ be the $d$-dimensional hypercube, and let $\varphi$ be a partial $d$-edge coloring of $Q_d$. A color class is the set of edges receiving one fixed color, and an induced ma…

For $k\in\mathbb{N}$, let $Q_{2k+1}$ denote the $(2k+1)$-dimensional hypercube. A graph is path-pairable if every pairing of its vertices can be joined by pairwise edge-disjoint pa…

Average-degree rainbow path conjecture. If $G$ has average degree at least $d$, then every proper edge-coloring of $G$ contains a rainbow copy of the path on $d$ edges.

Bipartition-respecting packing coloring conjecture. For every $n\geq 0$,

Hypercube spectral conjecture. The simple random walk $P^s$ is unstable, all eigenvalues of the corresponding linearized curvature-flow matrix are real, and

Let the $n$-Roberts graph encode the facet adjacencies of the $n$-cube, and let a hinged spanning cycle be a spanning cycle of this graph whose corresponding $2n$ facet adjacencies…

Let $Q_n$ be the $n$-dimensional hypercube and let $F_{n+1}$ be the dual-cube of dimension $n+1$. Let $k$ be a positive integer. Dual-cube lifting conjecture. If $Q_n$ has $k$ comp…

Let $[k]^d$ be the $d$-dimensional grid, with $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$, and let $D(k,[k]^d)$ denote the maximum density of a configuration avoiding the releva…

Let $Q_d$ be the $d$-dimensional hypercube, and let $m(Q_d;4)$ denote the minimum cardinality of a set that percolates under the $4$-neighbour bootstrap process on $Q_d$. Exact-siz…

Let $Q_7$ be the 7-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_7,C_4)$ for the maximum number of edges in a subgraph of $Q_7$ containi…

Let $Q_8$ be the 8-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_8,C_4)$ for the maximum number of edges in a subgraph of $Q_8$ containi…

Casselgren, Markström and Pham's conjecture. If $n$ and $d$ are positive integers, and $\varphi$ is a proper edge-precoloring of $K_{n,n}^d$ with at most $nd-1$ precolored edges, t…

Brimkov's conjecture. For every integer $d\geq 2$,

Hung's conjecture. An $n$-dimensional augmented cube admits a Hamiltonian decomposition.

For a graph $G=(V,E)$, a multipacking is a set $M\subseteq V$ such that, for every vertex $v\in V$ and every integer $k\geq 1$, $|N_k[v]\cap M|\leq k$, where $N_k[v]$ is the set of…

For a graph $G$ and a graph $F$, let $\delta^*(G,F)$ be the maximum minimum degree of a subgraph $H\subset G$ that has a proper edge-coloring with no rainbow copy of $F$. Let $Q_n$…

For a graph $G$ and a graph $F$, let $\mathrm{ex^*}(G,F)$ be the maximum number of edges in a subgraph of $G$ that has a proper edge-coloring with no rainbow copy of $F$. The $n$-d…

Let $\chi_n$ denote the parameter defined in the paper for the $n$-dimensional hypercube. Growth and limit conjecture. The sequence $\chi_n$ satisfies both of the following propert…

A normalized Hadamard matrix of order $n$ is a Hadamard matrix whose normalization is as used in the source. For a non-negative integer $r$ and a matrix $A$, let $A_{-r}$ be the su…

Let $s(n;r)$ denote the size of the smallest facet in $\operatorname{\mathcal{VR}}(Q_n;r)$. The source proves $s(n;r)\leq2r$ when a Hadamard matrix of order $2r$ exists and…

Let $Q_n=\\{\pm1\\}^n$ be the Hamming cube with Hamming distance. A subset is maximally diameter-$r$ if its diameter is $r$ and adjoining any point outside it produces diameter gre…

Let $G$ be a graph without a component isomorphic to $K_2$. For each $1\le i\le\Delta(G)$, let $n_i$ be the number of vertices of degree $i$, and define … The specific sequence irr…