Not to belabor a point that has already been beaten to death, but...

If a passenger is ASKED to pay the difference between coach and first class and agrees to do so, then obviously he or she is obligated to pay (because both parties agreed to this). My earlier point was that a passenger can't be "ambushed" by a charge posted to his or her account without such an agreement. I don't believe that any airline would do this, but if one did, I don't believe that the passenger would be required to pay. At a minimum, the airline has an obligation to ask the passenger to move; if the passenger refuses to do so, then all kinds of outcomes are possible, including arrest.

Bruce