Phantom maps and chromatic phantom maps

In the first part, we determine conditions on spectra X and Y under which e ither every map from X to Y is phantom, or no nonzero maps are. We also add ress the question of whether such all or nothing behavior is preserved when X is replaced with V boolean AND X for V finite. In the second part, we in troduce chromatic phantom maps. A map is n-phantom if it is null when restr icted to finite spectra of type at least n. We define divisibility and fini te type conditions which are suitable for studying n-phantom maps. We show that the duality functor Wn-1 defined by Mahowald and Rezk is the analog of Brown-Comenetz duality for chromatic phantom maps, and give conditions und er which the natural map Y --> (Wn-1Y)-Y-2 is an isomorphism.