Uniformly bounded information and social choice

The set of alternatives is infinite. If social welfare function f satisfies the Pareto criterion and there is a positive integer β such that for each pair of alternatives, {a,b}, there exists a set S(a,b) of at most β alternatives such that a and b can be socially ordered on the basis of the profile of preferences restricted to just S(a,b), then there are arbitrarily large finite subsets of X within which f is dictatorial on a rich sub-domain of profiles.