Let us consider a $n\times n$ grid. We colour each point of the grid using two colours.
We say that a grid is bad is there exists a set of 4 points on the grid with same color which are the vertices a square whose sides are parallel to the sides of the grid.
A grid which is not bad is said to be good.
Example: let say we colour using Red (R) or Blue (B).
The $3\times 3$ grid $\begin{matrix} R & B & R\cr B & R & B \cr R & B & B \end{matrix}$ is good.
A theorem of Furstenberg and Weiss states that there exists $n\geq 0$ such that any $(n+1)\times (n+1)$ grid is bad.
The natural question which arises is :
Question. What is the minimal value of $n$ ? In other words, what is the value of the unique integer such that there is a $n\times n$ good grid, but there is no $(n+1)\times (n+1)$ good grid ?
I don't know the answer, nor any estimate on the size of $n$. For the moment, the only method I coud think of to solve the problem is to use a computer.
The only small thing I know is that there is a $5\times 5$ good grid, so $n\geq 5.$
For example, the following grid is good:
$$\begin{matrix} R & R & R & R & R\cr B & R & B & R & B \cr B & R & B & B & R \cr B & B & R & R & B \cr R & B & B & R & B \end{matrix}$$
Thanks in advance for your answers!