  Julia Robinson has given a first-order definition of the rational integers
$\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall \exists
\forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8
variables, and where F is a polynomial.
  We show that for a large class of number fields, not including $\mathbb Q$,
for every $\epsilon>0$, there exists a set of primes $\cal S$ of natural
density exceeding $1-\epsilon$, such that $\mathbb Z$ can be defined as a
subset of the ``large'' subring $$\{x \in K : \ord_{\mathfrak p}x >0, \forall
\mathfrak p \not \in \cal S \}$$ of K by a formula of the form $(\exists
\forall \exists)(F=0)$ where there is only one $\forall$-quantifier, and where
F is a polynomial.
