Using Boolean Computation to Solve Some Problems from Ramsey Theory

Abstract

Using some examples from Ramsey theory, this article shows how to use Mathematicaʼs Boolean computational capability. ‡ Introduction 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]. ‡ Boolean RepresentationWe only concern ourselves with Boolean formulas in conjunctive normal form (cnf). A cnf is a conjunction of disjunctions of statements or their negations; the disjuncts are often called clauses.An example of a cnf is HA Í Ÿ B Í CL Ï HŸ A Í B Í CL Ï HŸ A Í Ÿ B Í Ÿ CL; letters represent statements and the symbols fi, fl, and Ÿ stand for "or," "and," and "not." A propositional formula is satisfiable if there is an assignment of true and false to its statements that makes the formula true when evaluated using the usual truth table rules.The Mathematica Journal 15

Computation of new diagonal graph Ramsey numbers

“…Cowen [9] used Mathematica's Boolean computational abilities to investigate some questions from graph Ramsey theory. In particular, he converted the problem of calculating R(K s , K t ) into a satisfiabilty problem.…”

“…blue[{2, 4}] || ! blue[{3, 4}])terminated.In[9], Cowen explored R(G 1 , G 2 , G 3 ), where G i were complete graphs. We ask the natural question "Is it feasible to adapt Cowen's approach to compute new diagonal graph Ramsey numbers R(G, G), where G is any simple connected graph?…”

Deleting Edges from Ramsey Minimal Examples

“…Utilizing Mathematica' s Boolean capability I was able to show that in several other cases the answer also was e=1(some of these examples appear in [1]). I began to conjecture that this might always be true, but didn't have any ideas on how to prove or disprove it nor did I know if anyone had already solved this problem.…”