75 problems

For a complete graph $K_r$, let $M_{K_r}(n)$ be the maximum running time of the $K_r$-bootstrap percolation process over all graphs on $n$ vertices. Bollobás–Przykucki–Riordan–Saha…

A $K$-fold Sidon set is a subset of $[N]$ containing no non-trivial solutions to equations of the form $\sum_{i=1}^4\alpha_i x_i=0$, where $\sum_{i=1}^4\alpha_i=0$ and the coeffici…

Let $\mathcal{U}$ be a $d$-dimensional update family, and let $p_c(\mathbb{Z}^d,\mathcal{U})$ denote its critical probability. The family $\mathcal{U}$ is subcritical when every he…

Let ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origi…

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…

Moment lower-bound conjecture. For any $r\geq 2$, the same inequality holds for every $\alpha\in(0,r-1]$. This conjecture concerns lower bounds for the critical probability of boot…

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 a base of Froböse percolation on a finite-size square mean a minimal configuration that percolates to that square under the rule that a point may be added when the other three…

Grippo–Pastine–Torres–Valencia-Pabon–Vera conjecture.

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 $C(k,l,m)$ be the cuboid consisting of $klm$ unit cubes, and call a configuration a solution when it has the dismantling property used throughout the paper. Small-cuboid existe…

Let $[n]^3$ be the $n\times n\times n$ cube, and let $s_3(n)$ denote the minimum possible area covered by the three orthogonal projections onto the faces of a solution to the faces…

Let $Q^n$ be the $n$-dimensional hypercube, and let $\lambda^*$ be a real number in $[-2,1/2]$. For … write $\Phi(p,Q^n)$ for the probability that random majority bootstrap percola…

Let $Q_n$ be the $n$-dimensional Hamming cube, and let $p_c(Q_n,n^a)$ denote the critical probability for the corresponding bootstrap percolation process with threshold $n^a$, wher…

Let $(\mathcal{K}_s,r_s)$ be a critical bootstrap percolation model in dimension $d$, let $A$ be the initially infected set, and let $\tau$ denote the critical time. Write…

Weak-saturation conjecture.

Consider a percolation process on a sufficiently regular rhombus tiling, with critical threshold $p_c$ and stable directions associated with the process. Stable-direction critical-…

Let $G=(V,E)$ be the adjacency graph of a uniformly repetitive (or uniformly recurrent) rhombus tiling. Let $F$ be a percolation process, monotone and freezing, on configurations…

Universality conjecture. The behaviour of majority bootstrap percolation is universal across all such graphs.

Let $\square_{i=1}^n K_k$ be the Cartesian product of $n$ copies of the complete graph $K_k$, and let $p_c(G,r)$ denote the critical probability for $r$-neighbor bootstrap percolat…

Let $\mathcal{U}$ be a subcritical update family on $\mathds{Z}^2$. Let $q_c(\mathcal{U})$ be its bootstrap-percolation critical probability, and let…

Let $G$ be a connected graph containing a percolating set, and suppose that the set $r$-percolates in $k$ rounds. For vertices $u,v$, let $D(u,v)$ be the length of a longest path b…

Let $\mathcal{G}$ be the collection of all connected graphs of order at least $3$ with at most two blocks. A graph is 2-bootstrap good if it contains at least one pair of vertices…

Let ${\mathcal U}$ be a critical two-dimensional update family. Its difficulty is … where a critical update family is balanced if there are no two opposite directions $v,-v\in S^1$…

For $d\geq4$, let $a_1,\ldots,a_d$ be side lengths of a $d$-dimensional grid, and let $m(a_1,\ldots,a_d;d)$ denote the minimum size of a percolating set for $d$-neighbour bootstrap…