The number of k-SAT functions

We study the number SAT(k; n) of Boolean functions of n variables that can be expressed by a k-SAT formula. Equivalently, we study the number of subsets of the n-cube 2(n) that can be represented as the union of (n - k)-subcubes. In The number of 2-SAT functions (Isr J Math, 133 (2003), 45-60) the authors and Imre Leader studied SAT(k; n) for k less than or equal to n/2, with emphasis on the case k = 2. Here, we prove bounds on SAT(k; n) for k greater than or equal to n/2; we see a variety of different types of behavior. (C) 2003 Wiley Periodicals, Inc.