A New Lower Bound for the Ramsey Number R(4, 8)

The lower bound for the classical Ramsey number R(4, 8) is improved from 56 to 58. The author has found a new edge coloring of K57 that has no complete graphs of order 4 in the first color, and no complete graphs of order 8 in the second color. The coloring was found using a SAT solver which is based on MiniSat and customized for solving Ramsey problems. Recently Exoo improved the lower bound for the classical Ramsey number R(4, 6) [Exoo 2012]. This note deals with a new lower bound for R(4, 8). The classical Ramsey number R(s, t) is the smallest integer n such that in any two-coloring of the edges of Kn there is a monochromatic copy of Ks in the first color or a monochromatic copy of Kt in the second color. Some of the interesting instances can be found at Exoo’s web site [Exoo] and at McKay’s web site [Mckay]. A recent summary of the state of the art for Ramsey numbers can be found in the Dynamic Survey [Radziszowski]. Exoo writes that some unsettled cases for two color classical Ramsey numbers such as R(4, 6), R(3, 10), and R(5, 5) can only be solved by using computer methods. The author think SAT solvers can be one of the promising tools to do this kind of work. Here we try to obtain a Ramsey graph R(s, t, n) [Mckay] by encoding the condition for R(s, t, n) to exist into a conjunctive normal form (CNF), called Ramsey clauses, as follows: