The theta-complete graph Ramsey number \(R(\theta_n, K_5) = 4n-3\) for \(n \geq 6\) and \(n \geq 10\).

Abstract

For any two graphs \(F_1\) and \(F_2\), the graph Ramsey number \(r(F_1, F_2)\) is the smallest positive integer \(N\) with the property that every graph of at least \(N\) vertices contains \(F_1\) or its complement contains \(F_2\) as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. In fact, we prove that \(r(\theta_n, K_5) = 4n-3\) for \(n \geq 6\) and \(n \geq 10\).