Let $A_{a, b} = \{ (x, y) \in \mathbb{Z}^2 : 1 \leq x \leq a, 1 \leq y \leq b \}$. Consider the following property, which we call Property R:

“If each of the points in $A$ is colored red, blue, or yellow, then there is a rectangle whose sides are parallel to the axes and vertices have the same color.”

Find the maximum of $|A_{a, b}|$ such that $A_{a, b}$ has Property R but $A_{a-1, b}$ and $A_{a, b-1}$ do not.