The class of decision problems for which the number of 'yes' instances of size n is upper-bounded by a polynomial in n. If SPARSE intersects NPC then P = NP [Mah82].
Contains TALLY.
The class that does not contain any languages. (It might not surprise you that I put this one in at the suggestion of a mathematician...)
Is the opposite of ALL, but does not equal the complement coALL = ALL.
Is closed under polynomial-time Turing reductions :-).
Equals SPARSE ∩ coSPARSE and TALLY ∩ coTALLY.
The class of decision problems for which every 'yes' instance has the form 0n (i.e. inputs are encoded in unary). If TALLY intersects NPC then P = NP [Mah82].
Contained in SPARSE.