Suppose there's an infinite chessboard with black and white queens, such that no black queen attacks a white queen. Also, the black queens and white queens have the same density $d$ (over large square areas, say). What's the maximum $d$?
I can see how to get $d = 1/8$: put black queens on $(4i, 2j)$ (for all integers $i$, $j$) and white queens on $(4i + 2, 2j + 1)$. Roughly speaking, the black and white queens stay a knight's move away from each other. I suspect that's the best.
Note that this is different from the problem where no queen can attack any other queen; here it's OK for a black queen to attack another black queen.