Linear Programming in Some Ramsey Problems

We derive new upper bounds for the classical two-color Ramsey numbers R(4, 5) ≤ 27, R(5, 5) ≤ 52, and R(4, 6) ≤ 43; the previous best upper bounds known for these numbers were 28, 53, and 44, respectively. The new bounds are obtained by solving large integer linear programs and with the help of other computer algorithms.