Let $\phi$ be a 3CNF formula with n variables and m clauses. A
simple nonconstructive argument shows that when m is
sufficiently large compared to n, most 3CNF formulas are not
satisfiable. It is an open question whether there is an efficient
refutation algorithm that for most such formulas proves that they
are not satisfiable. A possible approach to refute a formula
$\phi$ is: first, translate it into a graph $G_{\phi}$ using a
generic reduction from 3-SAT to max-IS, then bound the maximum
independent set of $G_{\phi}$ using the Lovasz $\vartheta$
function. If the $\vartheta$ function returns a value < m, this
is a certificate for the unsatisfiability of $\phi$. We show that
for random formulas with $m < n^{3/2 -o(1)}$ clauses, the above
approach fails, i.e. the $\vartheta$ function is likely to return
a value of m.