There are finitely many -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 k for each k > 2 . That is, we prove that, for any fixed m 1 > 2 , there are only finitely many cometric association schemes ( X , R ) with the property that the first idempotent in a Q -polynomial ordering has rank m 1 . 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. 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.