maps beginning with $v_0=p$: for example, the cofiber of the multiplication by $p$ map, denoted $S/p$, is a type one spectrum. At odd primes, we can continue this process and construct a type one spectrum $V(1)$ as the cofiber of a periodic self map $v_1: \Sigma^{2p-2}S/p\rightarrow S/p$. This iterative procedure was originally studied by L. Smith and H. Toda, however, it does not continue to produce type $n$ spectra as L. Nave showed in [@MR2630045].

To phrase the red-shift conjecture, it is necessary to have a notion of “height" for spectra that are not finite cell $S$-modules. We recall a definition due to N. Baas, B. Dundas, and J. Rognes that provides a notion of height in this context. This uses the fact that every $p$-local finite cell $S$-module $V$ admits a $v_n$-self map for some $n$, a consequence of the periodicity theorem of E. Devinatz, M. Hopkins, and J. Smith [@MR960945].

Let $X$ be a spectrum and let $\mathcal{T}_X$ be the thick subcategory of finite $p$-local spectra $V$ such that $$\label{eq red shift} V\wedge X \rightarrow v_n^{- 1} V\wedge  X$$ is induces an isomorphism in homotopy groups in sufficiently high degrees. Then if $\mathcal{T}_X=\cC_{n-1}$, we say that $X$ has 