4 problems

Let $P(x,y)$ be a polynomial that bijectively maps the non-negative integer lattice $\mathbb{N}_0\times\mathbb{N}_0$ onto the non-negative integers $\mathbb{N}_0$. Fueter–Pólya con…

Let $S\subseteq\mathbb{Z}^k$ be a finite set, and let $V(S)$ denote the set of lattice points mutually visible from every point of $S$. Write $D(V(S))$ for its asymptotic density a…

Let $d\geq 2$, let $G=\mathbb{Z}^d$, and let $H$ be a percolating-everywhere subgraph. Let $\mathcal{U}(H)$ be the graph obtained by adding the open edges of Bernoulli bond percola…

Let $d\geq 2$ and consider Bernoulli bond percolation on the integer lattice $\mathbb{Z}^d$. Benjamini–Häggström–Schramm conjecture. With probability one at the critical point, the…