On the use of senders in generalized ramsey theory for graphs

Abstract If F , G and H are graphs, write F → ( G , H ) to mean that however the edges of F are colored red and blue, either the red (partial) subgraph contains a copy of G or the blue subgraph contains a copy of H . Many interesting questions exist concerning this relation, particularly involving the case in which F is minimal for this property. A useful tool for constructing graphs relevant to such questions, at least when G and H are 3-connected, is developed here, namely graphs called senders . These senders are used to prove a number of theorems about the class of minimal F , as well as various related results. For example, let each of G and H be 3-connected, or a triangle. Then there exists an α > 0 such that if n is sufficiently large, there are at least e α n log n nonisomorphic F such that F → ( G , H ) in a minimal way.