  The famous pinching problem says that on a compact simply connected
$n$-manifold if its sectional curvature satisfies $K_{min} > (1/4)K_{max} > 0$,
then the manifold is homeomorphic to the sphere. In [8, problem 12], S. T. Yau
proposed the following problem: If we replace $K_{max}$ by the scalar
curvature, can we deduce similar pinching theorems? In our present note we give
an answer to this question in dimension $n = 4.$}
