A graph \(G\) is said to arrow the graphs \(F\) and \(H\), written \(G \rightarrow (F, H)\), 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 \(G \rightarrow (F, H)\). 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 \(G \rightarrow (F, F)\), or simply \(G \rightarrow F\)? We call this set of graphs the down-arrow Ramsey set of \(G\), or \(\downarrow G\). 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.