Compressive strength is how much inward force a given area of material can withstand before failure. The force tries to compress the atoms closer together.
Mohs hardness is the difficulty of a material to be scratched. Abrasion resistance. However, when an object scratches another, there first is a point of contact where the harder material is forced into the surface of the other, and then lateral forces shear off a grove from the softer material. But at this point of contact, the forces are compressive, trying to force the atoms together, until one of the materials give way.
As such, I would expect materials with a high Mohs hardness to have a high compressive strength and vice versa. This holds for ceramics such as silicon carbide, aluminum oxide, and others which have both a high compressive strength and a high Mohs hardness.
Does this trend hold for most/all materials and to be expected for the above reasons? Are there any counterexamples? If so, what could explain a high Mohs hardness and low compressive strength or vice versa?
