Which is distinct from a "totally random hash function", which is a hash function for which the hash associated with every value is uniformly randomly selected

Which is distinct from a "totally random hash function", which is a hash function for which the hash associated with every value is uniformly randomly selected from the key space. Totally random hash functions have very good properties, but they take a lot of space to store. (Exercise for the reader: if the hash function is {0,1}^n -> {0,1}^m (so the input is an n-bit string, and the output is an m-bit string), how much space do you need? Why isn't this compressible?)