No, the ice would not turn red in the scenario you described. When two substances are in thermal equilibrium, it means that they are at the same temperature and there is no net transfer of heat between them. In this case, the ice and water would reach the same temperature, but the color of the water (red dye) would not transfer to the ice.
The dye in the water is a solute, and when the water freezes into ice, it forms a solid phase while the dye remains dissolved in the remaining liquid water. The process of freezing separates the solute (dye) from the solvent (water) as the water molecules arrange themselves into a solid lattice structure.
If you were to melt the red ice, the dye would mix back into the water, and the resulting liquid would have a red color. However, during the freezing process, the dye does not migrate into the ice crystals, so the ice would remain colorless.
It's important to note that if you were to add dye directly to the water after the ice had already formed, the dye could potentially spread and color the ice as it melts and refreezes. But in the scenario of perfect thermal equilibrium with pre-existing ice and water, the ice itself would not turn red.