Partition relations for denumerable ordinals

AbstractThe principal result of this paper provides a nearly complete answer to the following question. For which integers k, m, n is it true that whenever we have a graph G whose set of vertices is an ordered set of order type ωn, then either this graph contains a complete subgraph with k vertices or the complement of G, Ḡ contains a complete subgraph whose set of vertices has order type ωm