On Some Small Classical Ramsey Numbers

This note is a report on a computer investigation of some small classical Ramsey numbers. We establish new lower bounds for the classical Ramsey numbers R(3, 11) and R(4, 8). In the first case, the bound is improved from 46 (a record that had stood for 46 years) to 47; and in the second case the bound is improved from 57 to 58. The classical Ramsey number R(s, t) is the smallest integer n such that in any twocoloring 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. A comprehensive summary of the current state of the art can be found in the dynamic survey on Small Ramsey Numbers [10]. In this note we present constructions that improve the lower bounds for the Ramsey numbers R(3, 11) and R(4, 8), and then describe the the modification in our search algorithm that led to the improvement in the R(3, 11) bound. Listings for these colorings are given at the end of this paper, and can also be found at the authors web site [4].