The generic existence of a core for q -rules (*)

A q-rule is where a winning coalition has q or more of the n voters. It is important to understand when, generically, core points exist; that is, when does the core exist in other than highly contrived settings? As known, the answer depends upon the dimension of issue space. McKelvey and Schofield found bounds on these dimensions, but Banks found a subtle, critical error in their proofs. The sharp dimensional values along with results about the structure of the core are derived here. It is interesting how these dimensional values correspond to the number of issues that are needed to lure previously supporting voters into a new coalition.