Uniformly bounded sufficient sets and quasitransitive social choice

X is infinite and social preference is quasitransitive. Subset Y of X is sufficient for {x,y} if x and y can be socially ordered with individual preference information over Y alone. If there is an integer β such that every pair of alternatives has a sufficient set with at most β members then for arbitrarily large finite subsets of X there is a rich subdomain of profiles within which a reduction in the amount of veto power must be accompanied by an equal increase in the fraction of pairs that are restricted, in that strict social preference prevails in only one direction.