Towards a theory of negative dependence

The FKG theorem says that the positive lattice condition, an easily checkab le hypothesis which holds for many natural families of events, implies posi tive association, a very useful property. Thus there is a natural and usefu l theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few p ossible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems. (C) 2000 American Institute of Physics. [S0022-2488(00)01703-5].