Checkerboard-Coloring $\mathbb{Z}^2$ 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 $n$:
$$
r_0=\frac{1}{4},\quad
r_1=\frac{1}{3},\quad
r_2=\frac{1}{2},\quad
r_3=\frac{1}{2},\quad
r_4\geq\frac{3}{5},\quad
r_5=\frac{9}{13},\quad
r_6=\frac{4}{5},\quad
r_7=\frac{8}{9},\quad
r_8=1
$$
 A: You can do $r_4$ with a $\frac{3}{5}$ ratio

. . X X X . . X X X
X X . . X X X . . X
. X X X . . X X X .
X . . X X X . . X X
X X X . . X X X . .
. . X X X . . X X X
X X . . X X X . . X
. X X X . . X X X .
X . . X X X . . X X
X X X . . X X X . .

This is an answer so that I can write the ascii art.
A: You may be able to get more bounds (upper and/or lower) by looking at this as a coloring of a graph and by thinking about how the relative proportions of black-black, black-white, and white-white edges relate to the proportions of black and white vertices.  The proportion of black vertices $p_b$ should be $p_{bb} + \frac{1}{2} p_{bw}$.
For $r_7$ each black vertex must be adjacent to at most $7$ black vertices, so each black vertex must be adjacent to at least one white vertex, so each black vertex must be associated with at least one black-white edge.  If every other edge is black-black then $\frac{7}{9}$ of the edges are black-black and $\frac{2}{9}$ of the edges are black-white, so the upper bound on the proportion of black vertices would be $\frac{7}{9} + \frac{1}{2} \cdot \frac{2}{9} = \frac{8}{9}.$  Therefore $\frac{8}{9}$ is both an upper and lower bound on $r_7$.  Maybe a similar approach could give upper $r_k$ bounds for more interesting $k$.
Edit: this answer is not so useful now that we have found a paper that gives exact values for all $k$ except $k=4$.
