  We prove that for any number $r$ in $[2,3]$, there are spin (resp. non-spin
minimal) simply connected complex surfaces of general type $X$ with
$c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In particular, this shows the
existence of simply connected surfaces of general type arbitrarily close to the
Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r \in [1,3]$
and any integer $q\geq 0$, there are minimal complex surfaces of general type
$X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$, and $\pi_1(X)$ isomorphic
to the fundamental group of a compact Riemann surface of genus $q$.
