What is a desert with a big cave?

yes there are caves in deserts

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Depending on the specific desert's geology. You may find rock-shelters eroded in sandstane by wind-blown sand, but not necessarily karst caves.

One desert that does hold caves is the Nullarbor Plain (S. Australia), with major, water-filled passages in limestone at some depth below the arid land surface.

There also caves in near-arid spots like the mountains of Iran.