What is the ruling on a husband and wife engaging in sexual intercourse in Ramadan a short time before the Fajr adhan, believing that the adhan had not yet occurred, and is expiation (kaffarah) required of them in this case, knowing that they were traveling and did not yet know the prayer times?

There is no blame on someone who consumes something that breaks the fast, thinking that dawn has not yet broken, only to find out later that it had. This is the more preponderant of the two scholarly opinions. For things that break the fast only invalidate it under three conditions: that one is knowledgeable, mindful, and acting of their own free will. Therefore, whoever does something that breaks the fast out of ignorance of the time or the ruling, there is no blame on them, and their fast is complete and valid. If the fast was broken by intercourse due to a mistaken assumption, then the person only has to make up that day.