There are finitely many QQ-polynomial association schemes with given first multiplicity at least three

In this paper, we will prove a result which is formally dual to the long-standing conjecture of Bannai and Ito which claims that there are only finitely many distance-regular graphs of valency kk for each k>2k>2. That is, we prove that, for any fixed m1>2m1>2, there are only finitely many cometric association schemes (X,R)(X,R) with the property that the first idempotent in a QQ-polynomial ordering has rank m1m1. As a key preliminary result, we show that the splitting field of any such association scheme is at most a degree two extension of the rationals.All of the tools involved in the proof are fairly elementary yet have wide applicability. To indicate this, a more general theorem is proved here with the result alluded to in the title appearing as a corollary to this theorem.