Oh yes, you are right. I'll edit, thanks!

ok I'll move it there, thank you!

But how would the $BP$-homology of $\Sigma X_n$ look like? Do we have any tool to find out? Thanks

Thank you again! Last question, promise: Do you think that is there a way to define these spectra such that this cofiber has at least one non torsion element? What i can think is a way to make $BP_k(X_n) \neq 0$ for almost every $n$ by having a double-indexed sum $\bigvee X_{n,i}$ such that $BP_*(X_{n,i})= \Sigma^{d_n} BP_* / (V_0^{k_{0,n}}, \dots, v_n^{k_{n,n}})$, where every $k_j > i$

I've written my question in the wrong way. What i meant was: Is it true that for every $i \geq 0$ there exist a finite (and then, with the proper $d_n,$ suspension) spectrum $X_i$ with the properties you wrote in the "ultrafilter" case, so with the $k_{n,i}$ going to infinity?

Thank you very much for the clear answer! Looks that this ultrafilter argument may be what i'm looking for, could you please suggest me a text-paper-book where to find something more about it? For example, the spectrum you mentioned with the $k_{n,i}$ going to infinity, is still finite and suspension? Thanks!!