Changing Views of Ramsey Numbers

In a red-blue coloring of a graph \(G\), every edge of \(G\) is colored red or blue. For two graphs \(F\) and \(H\), the Ramsey number \(R(F, H)\) of \(F\) and \(H\) is the smallest positive integer \(n\) such that every red-blue coloring of the complete graph \(K_n\) of order \(n\) results in either a subgraph isomorphic to \(F\) all of whose edges are colored red or a subgraph isomorphic to \(H\) all of whose edges are colored blue. While the study of Ramsey numbers has been a popular area of research in graph theory, over the years a number of variations of Ramsey numbers have been introduced. We look at several of these, with special emphasis on some of those introduced more recently.