Let $T$ be a complete first-order theory. Recall that a formula $\phi(\overline{x},\overline{y})$ has the *finite cover property* (fcp) if for all $n$, there exist $\overline{a}\_1,\dots,\overline{a}\_n$ such that $$T\models \lnot\exists \overline{x} \bigwedge\_{i = 1}^n \phi(\overline{x},\overline{a}\_i),$$
but for all $1\leq j \leq n$, $$T\models \exists \overline{x} \bigwedge\_{i \neq j} \phi(\overline{x},\overline{a}\_i).$$


As usual, a theory has nfcp if no formula has fcp. This condition was introduced by Keisler (in the context of Keisler's order) and studied by Shelah in *Classification Theory*, where it is shown that nfcp implies stability.


It is also easy to see that if $T$ has nfcp, then $T$ eliminates the quantifer $\exists^\infty$ (there exist infinitely many). That is, for every formula $\phi(\overline{x},\overline{y})$, there is a natural number $n$ such that for all $\overline{a}$, if $|\phi(M,\overline{a})|>n$, then $|\phi(M,\overline{a})|$ is infinite.


Now I have heard that there is a converse to these two facts. In particular, a partial converse (with the additional assumption of elimination of imaginaries) is stated as Theorem 2.8 in [this paper](http://arxiv.org/pdf/1312.0586v1.pdf), without proof or a reference:


**Theorem:** If $T=T^{eq}$ is stable and eliminates $\exists^\infty$, then $T$ has nfcp.


Can anyone provide a proof or a reference? The issue doesn't appear to be addressed in *Classification Theory*.



Is nfcp equivalent to stable + eliminates $\exists^\infty$?
In fact, this theorem is from Classification Theory: see Theorem 4.4 there (the "f.c.p. theorem").


Assertion (8) exactly says that $T^{eq}$ does not eliminate $\exists^{\infty}$ and the theorem says that for stable $T$, this is equivalent to having the f.c.p.


