Theorem 13. Assuming that:

$A \subset {[k]}^{n}$
$| A | \leq \frac{k^{n}}{2}$

Then

| \partial A | \geq \min {d | A |^{1 - \frac{1}{d}} k^{\frac{n}{d} - 1} : 1 \leq d \leq n} .