Hi, I'm running a 100 amp sub panel aproximatly 85 feet from the 200 amp main for an apartment. What size wire should I run?

In most States, you must be a licensed electrician to do work on multi-family dwellings.

Also, if you are using a panelboard for an apartment you cannot use that panelboard to feed devices that are used for the apartment complex, such as laundry rooms, lighting. That would have the renters paying for devices that belong to the complex.

As to your question about cable size. If the panel boards are listed for 60 (C), you must use a #1 AWG conductors. If your panel boards are listed for 75(C) than you can use a cable with #3 conductors.

Also you must use a four conductor cable that is listed for the type of installation you are going to use. (I.E. Below ground / Above ground)