Target Graphs and Ramsey Numbers

A sequence G1, G2, …, Gk of pairwise edge-disjoint monochromatic subgraphs of a graph G with a red-blue edge colouring is a Ramsey chain in G if Gi has i edges for 1≤i≤k and Gi is isomorphic to a subgraph of Gi+1 for 1≤i≤k−1. The subgraphs Gi are the links of the Ramsey chain and the terminal subgraph Gk of size k is the target link of the chain. A graph H without isolated vertices is a target graph if there exists a positive integer n such that every red-blue colouring of Kn results in a Ramsey chain with target link H. For a target graph H, the target Ramsey number TR(H) of H is the minimum positive integer n such that for every red-blue colouring of Kn, there exists a Ramsey chain having H as its target link. It is shown that TR(H) exists for every graph H without isolated vertices and for the standard (diagonal) Ramsey number R(H)=R(H,H), it follows that TR(H)≥R(H). The number TR(H) is determined for several graphs H and it is shown that TR(H)=R(H) for all these graphs H.