2 problems

For each positive integer $k$, let $\Delta_k(n)$ denote the maximum upper density of a subset $S\subseteq\mathbb{Z}^k$ whose maximum degree is at most $n$, where degree counts neig…

Conjecture A. If $d(S)\leq 3$, then $S$ has upper density at most $1/2$ in $L$. This is presented as a stronger version of the still-life density problem: patterns with density nea…