  We show that there is $n\in \mathbf N$, a finite system $\Sigma(\vec x,\vec
y)$ of equations and inequations having a solution in some group, where $\vec
x$ has length $n$, and $\epsilon>0$ such that: for any group $G$ and any $\vec
a\in G^n$, if the system $\Sigma(\vec a,\vec y)$ has a solution in $G$, then
there is no $(\vec a,\epsilon)$-F{\o} lner set in $G$. The proof uses ideas
from model-theoretic forcing together with the observation that no amenable
group can be existentially closed. Along the way, we also observe that no
existentially closed group can be exact, have the Haagerup property, or have
property (T). Finally, we show that, for $n$ large enough and for $\epsilon$
small enough, the existence of $(F,\epsilon)$-F{\o} lner sets, where $F$ has
size at most $n$, cannot be expressed in a first-order way uniformly in all
groups.
