Almost every 2-SAT function is unate

Bollobás, Brightwell and Leader [2] showed that there are at most 2( n 2)+o(n 2) 2-SAT functions on n variables, and conjectured that in fact almost every 2-SAT function is unate: i.e., has a 2-SAT formula in which no variable’s positive and negative literals both appear. We prove their conjecture, finding the number of 2-SAT functions on n variables to be 2( n 2)+n(1 + o(1)). As a corollary of this, we also find the average number of satisfying assignments of a 2-SAT function on n variables. We also find the next largest class of 2-SAT functions, and show that if k = k(n) is any function with k(n) < n 1 4 for all sufficiently large n, then the class of 2-SAT functions on n variables which cannot be made unate by removing 25k variables is smaller than 2( n 2)+n−kn for all sufficiently large n.