23 problems

Develin–Hartke conjecture. There exists an outbreak on $\mathbb{Z}^d$ which cannot be contained by deploying $f(n)$ firefighters at time $n$.

Prime divisibility embedding conjecture. There is a Euclidean motion $T$ such that

Divisibility embedding conjecture. There is a set $N\subset\mathbb{Z}^n$ such that $k\cdot N$ is congruent to $M$.

Embedding conjecture. One can find a Euclidean motion $T$ such that

Let $\mathbb{Z}^n$ have the Manhattan metric, let $\{0,1\}^n\subseteq\mathbb{Z}^n$ have the induced metric, and let $\widetilde{H}_i(-;\mathbb{Z})$ denote reduced homology with int…

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

Let $S\subseteq\mathbb{N}^2$ contain no three points on a common line. No-three-in-line density conjecture. … Although finite $n\times n$ configurations have the elementary upper b…

Let $\alpha,n,d\in\mathbb{Z}^{+}$, let $C\subset\mathbb{Z}^{d}$ satisfy $|C|=n$, and let $\mathcal{T}(\alpha,C)$ denote the set of canonical translational tilings of $C$ by tiles o…

Let $d\geq 1$ and $p\in(0,1)$. On the graph $\mathbbm{Z}^d$, whose vertices at Euclidean distance $1$ are joined by an edge, use the componentwise order $u\leq v$ defined by…

Weakly periodic tiling conjecture. There exists a weakly periodic $F$-tiling of $\mathbb Z^d$; more precisely, there exists an $F$-tiling $T\subseteq\mathbb Z^d$ admitting such a f…

For $d\geq 3$, let $p_{1,c}(\mathbb{Z}^d)$ denote the critical probability for $1$-independent percolation on the integer lattice. High-dimensional percolation conjecture. There ex…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Let $\operatorname{MD}_{\mathrm{coins}}$ denote the coin-valued median dynamics process on $\mathbb{Z}^d$, and let $\eta_t(\underline{0})$ be the opinion at the origin. Coin-median…

Consider median dynamics on $\mathbb{Z}^2$, with $\eta_\infty(0)$ denoting the limiting opinion at the origin. Stable-structure tail conjecture. … The exponent $4$ comes from the s…

Let $p_c(G)$ be the infimum of the densities $p\in[0,1]$ for which zero-temperature Glauber dynamics on $G$, started from i.i.d. $\operatorname{Bernoulli}(p)$ opinions, converges a…

Balog–Shakan conjecture. One has

Let $cmathcal{E}_n$ be the external DLA cluster on $cmathbb{Z}^d$ started from $cmathcal{E}_0=\{0\}$, and let $r(cmathcal{E}_n)$ denote its radius. External DLA radius-growth conje…

A tile is a finite non-empty subset of $\mathbb{Z}$, and a tile $T$ tiles $\mathbb{Z}^d$ if $\mathbb{Z}^d$ can be partitioned into translates, rotations, and reflections of $T$. Gr…

Classification conjecture. If $\mathcal{T}$ is a primitive, regular, hexagonal tiling of $\mathbb{Z}^2$ by $90^\circ$-symmetric tiles, then

Let $m$ be a positive integer, and let $\chi(\mathbb{Z}^{3},\sqrt{m})$ denote the minimum number of colours needed to colour $\mathbb{Z}^{3}$ so that points at distance $\sqrt{m}$…

Let $\cal C$ be a clutter with incidence vectors $v_1,\dots,v_q$, and suppose every minor $\cal C'$ satisfies $\tau({\cal C}')=\nu({\cal C}')$. If $x^{v_1},\dots,x^{v_q}$ have degr…

An $n_m$-cluster is an integral point set in general position consisting of $n$ points in the integer grid $\mathbb{Z}^m$. Erdős–Noll conjecture. For any $m>1$ and $n>1$, there exi…

Let $\mathfrak{S}$ be a peg-solitaire board, let $\mathscr{D}(\mathfrak{S})$ be its set of legal-move vectors, and let $\mathcal{B}\subset\mathscr{D}(\mathfrak{S})$ be a basis of…