The Down-Arrow Ramsey Set of a Graph

A graph G is said to arrow the graphs F and H , written G → ( F , H ) , if every red-blue coloring of G results in a red F or a blue H . The primary question has been determining graphs G for which G → ( F , H ) . If we consider the version for which F = H , then we can ask a different question: Given a graph G , can we determine all graphs F such that G → ( F , F ) , or simply G → F ? We call this set of graphs the down-arrow Ramsey set of G , or ↓ G . The down-arrow Ramsey set of cycles, paths, and small complete graphs are determined. Graph ideals and graph intersections are introduced and a method for computing down-arrow Ramsey sets is described.