Sure, Ramsey's Theorem guarantees that Ramsey numbers exist. But at no point is it assumed that the graph is connected. The section you linked puts it quite sim

Sure, Ramsey's Theorem guarantees that Ramsey numbers exist. But at no point is it assumed that the graph is connected. The section you linked puts it quite simply: > The Ramsey number is the minimum number of vertices, v = R(m, n), such that all undirected simple graphs of order v, contain a clique of order m, or an independent set of order n. The graph is undirected and simple. But it need not be connected.