Problem of the week

2019-13 Property R

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.