The essentially free spectrum of a variety

We partially prove a conjecture from [MeSh] which says that the spectrum of almost free, essentially free, non-free algebras in a variety is either empty or consists of the class of all successor cardinals. Introduction and notation Suppose that T is a variety in a countable vocabulary τ . This means that τ is a countable set of function symbols and T is a set of equations, i.e. sentences of the form ∀x1, . . . , xn (σ1(x1, . . . , xn) = σ2(x1, . . . , xn)) where σi are τ−terms. The class of all 1 The author is supported by the NSERC 2 The author is supported by the United States–Israel Binational Science Foundation; publication 417. 3 The author is supported by the Schweizer Nationalfonds.