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.

The best solution was submitted by 하석민 (수리과학과 2017학번). Congratulations!

Here is his solution of problem 2019-13.

An incorrect solution was received. Late solutions are not graded.