The ultra‐weak Ash conjecture and some particular cases

Ash's functions N _{σ ,k} count the number of k ‐equivalence classes of σ ‐structures of size n . Some conditions on their asymptotic behavior imply the long standing spectrum conjecture. We present a new condition which is equivalent to this conjecture and we discriminate some easy and difficult particular cases. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)