  Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$
endowed with a stable vector bundle $V$, usually lead to an anomaly mismatch
between $c_2(V)$ and $c_2(X)$; this leads to the question whether the
difference can be realized by a further bundle in the hidden sector. In
math.AG/0604597 a conjecture is stated which gives sufficient conditions on
cohomology classes on $X$ to be realized as the Chern classes of a stable
reflexive sheaf $V$; a weak version of this conjecture predicts the existence
of such a $V$ if $c_2(V)$ is of a certain form. In this note we prove that on
elliptically fibered $X$ infinitely many cohomology classes $c\in H^4(X, {\bf
Z})$ exist which are of this form and for each of them a stable SU(n) vector
bundle with $c=c_2(V)$ exists.
