  This work concerns commutative algebras of the form $R=Q/I$, where $Q$ is a
standard graded polynomial ring and $I$ is a homogenous ideal in $Q$. It has
been proposed that when $R$ is Koszul the $i$th Betti number of $R$ over $Q$ is
at most $\binom gi$, where $g$ is the number of generators of $I$; in
particular, the projective dimension of $R$ over $Q$ is at most $g$. The main
result of this work settles this question, in the affirmative, when $g\le 3$.
