What is the ruling on lying about a past relationship, is expiation obligatory for it, and what should be done to correct the course after lying?

The oath you took that you "do not know" this person, based on your assumption, is an oath that submerges one in sin (yamin ghamus). It is called so because it submerges its taker in sin, and then in the Fire. It is considered one of the major sins. Expiation is not required for it, according to the preponderant opinion among scholars. Rather, what is obligatory is repentance to Allah Almighty, seeking forgiveness, feeling remorse, and resolving not to return to such an act.