On the complexity of division and set joins in the relational algebra

We show that any expression of the relational division operator in the relational algebra with union, difference, projection, selection, constant-tagging, and joins, must produce intermediate results of quadratic size. To prove this result, we show a dichotomy theorem about intermediate sizes of relational algebra expressions (they are either all linear, or at least one is quadratic), and we link linear relational algebra expressions to expressions using only semijoins instead of joins.