Ramsey numbers for theta graphs (Q666520)

Summary: The graph Ramsey number \(R(F_1, F_2)\) is the smallest integer \(N\) with the property that any complete graph of at least \(N\) vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to \(F_1\) all of whose edges are red or a subgraph isomorphic to \(F_2\) all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine \(R(\theta_4, \theta_k)\), \(R(\theta_5, \theta_k)\) for \(k \geq 4\). More specifically, we establish that \(R(\theta_4, \theta_k) = R(\theta_5, \theta_k) = 2k - 1\) for \(k \geq 7\). Furthermore, we determine \(R(\theta_n, \theta_n)\) for \(n \geq 5\). In fact, we establish that \(R(\theta_n, \theta_n) = (3n/2) - 1\) if \(n\) is even, \(2n - 1\) if \(n\) is odd.