I truly believe this is a materials quality problem and not so much a chemistry problem. Not all travertine is the same, it's quality depend A LOT of where it was mined from and how it was formed. Travertine is nothing more than limestone that has formed from a flowing water source. True travertine comes from hot spring sources where superheated water that is saturated with calcium carbonate rises up to the surface. Once there, the excess dissolved CO2 rapidly outgassed and the pH rises. As the pH rises the calcium carbonate precipitates out of solution. There are two forms of calcium carbonate - calcite and aragonite (different crystalline structures). If the water that the travertine deposit forms in is hot (the hotter the better), then the aragonite is the preferred precipitate and the travertine will be harder, more dense and less porous. If the water is cooler, then the calcite form is favored where the travertine will be less dense, softer and more porous. In the extreme case of fluvial waters (free flowing cool waters like rivers and lakes), the deposit that is formed is very porous and weak; as such, it is not technically travertine but it is called tufa.
So, my gut feeling is that the homeowner simply has a poor quality batch of travertine that is likely more porous and soft. As such, the constant exposure to water, wet/dry cycling and chloride exposure (from the chlorine used in the pool) has simply caused the travertine to spall and disintegrate. Chloride and sulfate attack are well known problems in cementitious materials especially in arid climates where the rapid evaporation allows the pool water to deposit lots of chloride into the surface of the stone. Those tiles should have been sealed to prevent the damage you are seeing. My guess is they were not to prevent them from being slippery.
From the homeowner:
"Travertine pavers are very popular outside and absolutely need maintenance and proper sealing to stay solid and last. Lime content inside the pavers will turn corrosive when wet, (pool runoff, sprinklers, rain, fountains, drip systems, etc...) The lime can even spread from pavers to pavers like a disease." Very specific special penetrating sealer is needed to neutralize the lime. So, it does not seem like the chemical content of the pool water is what has caused it."