We'll call this one "The Case of the Incredible Shrinking (and Curling) Floor". To coin a phrase - you hit the nail right on the head - a cold basement with a hot furnace and we're willing to bet that there's no floor insulation either.

First, it will be important to eliminate the real cause of the damage to prevent a recurrence once the floor tiles have been repaired. This is done by reducing the transmission of moist air that is currently traveling from the basement to the kitchen via the floor. All you have to do is fill the basement-ceiling cavity (below the kitchen floor) with spun glass insulation. Use the paper-backed type and make sure the paper side is placed up toward the kitchen. Both the insulation and its backing will absorb a substantial amount of the moisture giving natural air circulation a chance to evaporate the moisture at the insulation level instead of the finish floor covering level. Also, the addition of the insulation will make that area above a more comfortable and energy efficient place to live. Insulation netting is stapled to the floor-framing members (below the insulation) to hold the material in place.

To repair the curling tiles they must be made soft and pliable. Heating them does this. Although a hair dryer does work, we prefer to use a clothes iron to a bath towel. Once the tile is soft, re-affix the curled edges to the floor with a generous portion of Dap Tub and Tile Caulk (a great all purpose glue). Once the edges have been rejoined, wipe off the excess adhesive with a damp towel. Finally, cover the repaired seam with wax paper and a heavy object to hold everything in place. The wax paper will prevent the heavy object from sticking to the glued seam.

Our guess is the tiles that have an eighth-inch gap have shrunk. Under the conditions mentioned we aren't surprised. Here, the tiles will have to be replaced. Removing them will be easier if they are heated first. However, if replacement tiles aren't available, or too expensive, then fill the joint with ceramic tile grout (in a matching color). This will keep food and debris from getting into the joints, and help to hide the unsightly gap.

Germany or not, a landlord that isn't willing to pay for sensible repairs that improve the life, value and condition of the property is, in our opinion, a slum-lord. Move - and stick them with an empty property. Remember good old American supply vs. demand.