Lower Bounds on Ramsey Numbers \(R(6, 8), R(7,9)\) and \(R(8, 17)\)

Abstract

For integers \(s,t \geq 1\), the Ramsey number \(R(s, t)\) is defined to be the least positive integer \(n\) such that every graph on \(n\) vertices contains either a clique of order \(s\) or an independent set of order \(t\). In this note, we derive new lower bounds for the Ramsey numbers: \(R(6,8) \geq 129\), \(R(7,9) \geq 235\) and \(R(8,17) \geq 937\). The new bounds are obtained with a constructive method proposed by Xu and Xie et al. and the help of computer algorithm.