Changing Views of Ramsey Numbers

Abstract

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.