I was wondering whether there is an analogue result to the minimality of Wang tiling, in the direction of maximality.
I think that the paper by Jeandel and Rao, shows that the minimal number of Wang tiles to generate an aperiodic tiling of the plane is $11$. Is there a converse result, that given $k$ colors appearing on a set of Wang tiles, the Wang tiles can have at most $N(k)$ to be an aperiodic set of tiles?
If I am not mistaken, this question only makes since when $k\geq 4$. It also seems clear to me that $N(k)< k^4-4$ for example. I was wondering however if there are some more intelligent estimates known so far in the literature. I would be thankful if any one can point me to some results of this nature.
