Linear programming in some Ramsey problems (Q1328393)

By using some graphical algorithms and solving large integer linear programming problems the authors obtain the following new upper bounds for some classical Ramsey numbers, namely \(R(4,5) \leq 27\), \(R(5,5) \leq 52\), and \(R(4,6) \leq 43\). Their approach to the search is outlined in the paper. They also note in an addendum that by improving one of the graphical algorithms, they can now verify that \(R(4,5) = 25\), \(R(5,5) \leq 49\), and \(R(4,6) \leq 41\).