Using Boolean Computation to Solve Some Problems from Ramsey Theory

Mathematica’s industrial-strength Boolean computation capability is not used as often as it should be. There probably are several reasons for this lack of use, but it is our view that a primary reason is lack of experience in expressing mathematical problems in the form required for Boolean computation. We look at a typical problem that is susceptible to Boolean analysis and show how to translate it so that it can be tested for satisfiability with Mathematica’s built-in function SatisfiableQ. The problems we investigate come from an area of mathematics called Ramsey theory. Although Ramsey theory has been studied extensively for over 80 years and still provides many challenges, we neglect the theory (for the most part) and instead concentrate on translating the problems so that they are amenable to Boolean computation and then see what can be accomplished by computation alone. Those interested in learning a little more about Ramsey theory can consult [1]; for a standard reference, see [2].