That is the most ridiculous and erroneous information I have heard. Sometimes people are confused about travertine, but that info is just plain wrong.
Let's talk about travertine pavers, as that is what I would suggest. This is because they can be easily dry laid like concrete pavers. Travertine does not discolor. It is a natural stone that maintains its beauty. It doesn't matter if it is in the sun or shade. If you get a quality travertine, the holes will be minimal.
I have written a page on travertine, explaining the differences in various types and I discuss quality. Here is the link. Travertine Pavers
Travertine pavers are one of the nicest materials you can use around a pool. I never recommend slate or bluestone for pool paving since they get very hot.
Tumbled pavers can look nice too and typically less expensive, but travertine pavers are special. Please visit my page on Concrete Pavers