They absolutely should not be tiling directly on the wood floor. You need cement board or something like Ditra under the tile. You also need to make sure that your floor's under structure will support stone tile. There is a tile website with a tool that will help you figure out if the deflection in your floor will support the tile.

Terriks...thanks so much . TO be clearer, it seems to be a choice of floating vs. installing with backer board. My recent research seems to say floating is more difficult to install but best at preventing cracks due to any future shifting....and we do live in Ca. I 'm wondering if floating truly is the preferred method,

If you have the elevation to add the required thickness, floating a floor will give you a much more stable installation. Especially for 18" travertine.

If you do go with cement board, make sure they thinset the cement board down to the subfloor. The thinset prevents any vertical movement of the cement board by filling any gaps between the ply and the cement board. It's not really intended to bond them together, it's more of a gap-filler. The fasteners will bond them together.

Thinset between the cement board and the plywood is not optional. It's mandatory.

18" travertine is an animal all by itself. I'd much prefer that to go over a mud bed. If not, then you need to really look at the floor structure. I'd want two layers of plywood (3/4" subfloor covered with 3/8" or 1/2" ply underlayment) and then a membrane like Ditra instead of cement board.

The larger the tile, the better shape your floor needs to be. No humps and dips, it needs to be pretty flat or the larger tiles can rock over any humps and you;ll end up with lippage between adjacent tiles. When installing natural stone versus ceramic or porcelain, the better shape your floor needs to be. Natural stone requires a stronger floor with less flex, especially for travertine.

Installing 18" travertine directly to an existing plywood subfloor that was under carpet? Do that and you could very well end up with little bitty bits of travertine as the tile cracks over time.

If you can't get the large format 18" travertine installed correctly, it might be better to move to a smaller format natural travertine tile or to a porcelain travertine look-a-like.

Dear mongoct....thank you for such a wonderfully detailed reply. I printed it for our tile man to read. Not certain, but think he was wavering and is now going for the floating method. Cannot say thanks enough. Even made it much clearer to me!