  Let $I$ be an ideal generated by quadrics in a standard graded polynomial
ring $S$ over a field. A question of Avramov, Conca, and Iyengar asks whether
the Betti numbers of $R = S/I$ over $S$ can be bounded above by binomial
coefficients on the minimal number of generators of $I$ if $R$ is Koszul. This
question has been answered affirmatively for Koszul algebras defined by three
quadrics and Koszul almost complete intersections with any number of
generators. We give a strong affirmative answer to the above question in the
case of four quadrics by completely determining the Betti tables of height two
ideals of four quadrics defining Koszul algebras.
