Enumerating k-SAT functions

How many k-SAT functions on n boolean variables are there? What does a typical such function look like? Bollobás, Brightwell, and Leader conjectured that, for each fixed k ≥ 2, the number of k-SAT functions on n variables is (1 + o(1))2 ( n k )+n , or equivalently: a 1 -o(1) fraction of all k-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for k = 2. The conjecture was confirmed for k = 2 by Allen and k = 3 by Ilinca and Kahn.

We show that the problem of enumerating k-SAT functions is equivalent to a Turán density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollobás-Brightwell-Leader conjecture for k = 4 by solving the corresponding Turán density problem. Our solution applies a recent result of Füredi and Maleki on the minimum triangular edge density in a graph of given edge density. In an appendix (by Nitya Mani and Edward Yu), we further confirm the k = 5 case of the conjecture via a brute force computer search.