Green-Tao Numbers and SAT

We consider the links between Ramsey theory in the integers, based on van der Waerden’s theorem, and (boolean, CNF) SAT solving. We aim at using the problems from exact Ramsey theory, concerned with computing Ramsey-type numbers, as a rich source of test problems, where especially methods for solving hard problems can be developed. We start our investigations here by reviewing the known van der Waerden numbers, and we discuss directions in the parameter space where possibly the growth of van der Waerden numbers vdwm(k1, . . . , km) is only polynomial (this is important for obtaining feasible problem instances). We introduce transversal extensions as a natural way of constructing mixed parameter tuples (k1, . . . , km) for van-der-Waerden-like numbers N(k1, . . . , km), and we show that the growth of the associated numbers is guaranteed to be linear. Based on Green-Tao’s theorem (“the primes contain arbitrarily long arithmetic progressions”) we introduce the GreenTao numbers grtm(k1, . . . , km), which in a sense combine the strict structure of van der Waerden problems with the (pseudo-)randomness of the distribution of prime numbers. Using standard SAT solvers (look-ahead, conflict-driven, and local search) we determine the basic values. It turns out that already for this single form of Ramsey-type problems, when considering the best-performing solvers a wide variety of solver types is covered. For m > 2 the problems are non-boolean, and we introduce the generic translation scheme, which offers an infinite variety of translations (“encodings”) and covers the known methods. In most cases the special instance called nested translation proved to be far superior over its competitors (including the direct translation).