TITLE: Bicoloring of $\mathbb{N}^2$, avoiding set of patterns, is the maximal limit density rational?
QUESTION [15 upvotes]: Consider a bi-coloring of $\mathbb{N}^2$, (black and white), where we wish to maximize the limit (limsup) of the density of black squares in $[n] \times [n]$ as $n \to \infty$. Here, we identify each point with a square in the obvious way.
However, we have specified a finite set of black sub-patterns that are not allowed.
That is, a pattern is a subset of $[k]\times [k]$, and any translation of this subset is not allowed in the coloring as all black squares.
For example, perhaps we do not allow two adjacent black squares.
Then, a checkerboard coloring has limit density $1/2$.
Updated after answer
Q1: For any finite set of forbidden patterns, will the coloring(s) maximizing the density always be periodic? NO
Q2: Is there a finite set of forbidden patterns, such that the maximal limit density can only be obtained by a non-periodic coloring? YES
Q2b: Is there a finite set of forbidden patterns, such that the maximal limit density is an irrational number? Note that this requires a non-periodic pattern.
Q3: Is there a limit density that is not realizable, that is, there is a sequence of colorings, $c_1,c_2,\dots$ with increasing limit density, but the limit density is not realizable by any coloring?
Q4: Since this feels closely related to tiling problems and permutation patterns, might it be that the question "Does the set $F$ allow a coloring with limit density $\geq d$?" is undecidable?  YES 
According to bijection with Wang tiles, where all tiles having equal density.
Deciding if a finite set of Wang tiles, tile the plane, is undecidable.
*Note that for every set of forbidden patterns, coloring all squares white is a valid coloring. *
Note: We only consider a finite set of forbidden patterns in all questions.

REPLY [3 votes]: I believe the answer to Q2b is yes, there does exist setups where the limit density is irrational.
Consider the Kari-Culik (Wang) tilings, and let $\Phi$ be projection onto the top label of each tile (treating $0'$ as $0$).  Durand, Gamard, and Grandjean in http://arxiv.org/abs/1312.4126 show that for any Kari-Culik tiling $x$, the row averages of $\Phi(x)$ converge.  Further, if $(\gamma_i)_{i\in\mathbb Z}$ is a vector of row averages, 
$\gamma_i=f(\gamma_{i-1})$ where $f(t) = \begin{cases}2t&t\in[1/3,1)\\ t/3 & t\in[1,2]\end{cases}$.
$f$ is conjugate to an irrational rotation via the map $\varphi:[1/3,2]\to [0,1]$ given by $\varphi(t) = \frac{\log t+\log 3}{\log 6}$, and so since an irrational rotation gives a uniform orbit density, pushing forward through $\varphi$, we can compute the average
$\mathrm{Ave}(\Phi(x)) = \frac{5}{\log 216}$
for any Kari-Culik tiling $x$.  I didn't think about the correspondence of Wang tilings and your problem in detail, but after translating the Kari-Culik tiles into the framework of your bi-coloring, an irrational average should be preserved.