Some Ideals with Large Projective Dimension

For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of $I$ grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials.