Anyone know what happens if you buy a first class ticket for an airline, but
they have no first class seats available?

That would be quite surprising. As far as I know, airlines do not
overbook first class like they do coach.

prices!) left. Problem is their seating plan shows no seats available onm
one of the two legs and says "a seat will be assigned at check in".

All that means is that they're holding back seat assignments for
some purpose. I've been on plenty of flights where a few days ahead
the seat map claimed it was 100% full, and when we actually boarded they
had room for 50 standbys.

R's,
John