If every square of the unit square lattice in the plane is colored black or white according to a set of rules, is there a way to find the maximum asymptotic ratio $r_n$ of the number of black squares to the total number of squares?
If the rules are that each black square can have at most $n$ adjacent (side-by-side or corner-to-corner) black squares, here are some partial results for every possible $n$: $ r_0=\frac{1}{4},\quad r_1=\frac{1}{3},\quad r_2=\frac{1}{2},\quad r_3=\frac{1}{2},\quad r_4\geq\frac{3}{5},\quad r_5=\frac{9}{13},\quad r_6=\frac{4}{5},\quad r_7=\frac{8}{9},\quad r_8=1 $