Koszul Algebras Defined by Three Relations

This work concerns commutative algebras of the form $R=Q\mathrm{/}I$, 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 $\left(\genfrac{}{}{0px}{}{g}{i}\right)$, 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 $g\le 3$.