In an elastic collision between two identical balls, velocities are exchanged. Ball A starts at height 4h, so its speed before collision is $\sqrt{2g\times 4h}=\sqrt{8gh}$. Ball B starts at height h, so its speed is $\sqrt{2gh}$. After collision, A's speed becomes $\sqrt{2gh}$ and B's becomes $\sqrt{8gh}$. The maximum height reached is proportional to the square of the speed. Therefore, the ratio of heights is $\frac{{\left(\sqrt{2gh}\right)}^2}{{\left(\sqrt{8gh}\right)}^2}=\frac{2gh}{8gh}=\frac{1}{4}$.

Final Answer: 1 : 4