from AFAR....looks like a bit MORE than 3 episodes of 'dampness'with no accumulation.... from here.

can`t see the WALLS good enough to offer much with THOSE.

'Could' be partly or all, a problem UNDER the floor, could be from a back up in lateral line AND/OR ,if there is a sump pump...a problem in the tiles that lead-to sump pit OR, IF sump, problem(S) with the pump itself.

I have SELLER WRITE down something...be more CLEARER-honest and/or,, ask them to have an honest-experienced plumber SNAKE through any-a;; floor drains and through the storm trap IF there is one. And if there is a sump pump, have plumber snake through the tiles that empty into pit.

Take a good-long look at EVERY bsmt wall, check every rod hole, and see if there is a crack that may have been patched over,injected etc. You`ll want that repaired correctly from the outside, NOT any inside crap.

If there isn`t a sump pump(s), house 'MIGHT' need one, doesn`t mean it needs any inside drain tile or baseboard system.

Thats a few things i`d do/ask etc. If they aren`t willing to be more clearer then maybe find another home, there`s plenty out there.

Thanks for your thoughts on the photos. I forgot to mention that there are no similar colorations or evidence of water on the walls... this coloration is limited to the floor. Also, we have asked them to define "dampness" and "accumulation" as well.