affine hyperplane $H' \subset \C^d$ such that $|H' \cap V| \geq \delta \cdot n$.

Let $h \in \C^{d\ell}$ be the vector perpendicular to $H$ so that $x \in H$ iff ${\langle x,h \rangle}=b$ for some $b \in \C$. Observing the product structure of $\C^{d\ell} = (\C^d)^\ell$ we can write $h = (h_1,\ldots,h_\ell)$ with each $h_i \in \C^d$. W.l.o.g suppose that $h_1 \neq 0$. For each $a = (a_2,\ldots,a_\ell) \in V^{\ell-1}$ let $V^\ell_a = V \times \{a_2\} \times \ldots \{a_\ell\}$. Since there are $n^{\ell-1}$ different choices for $a \in V^{\ell-1}$, and since $$|V^\ell \cap H| = \sum_{a \in V^{\ell-1}}|V^\ell_a \cap H|,$$ there must be some $a$ with $|V^\ell_a \cap H| \geq \delta \cdot n$. Let $H' \subset \C^d$ be the hyperplane defined by the equation $$x \in H' \,\,\text{iff}\,\, {\langle x,h_1 \rangle}+ {\langle a_2,h_2 \rangle} + \ldots + {\langle a_\ell,h_\ell \rangle} = b.$$ Then, $H' \cap V$ is in one-to-one correspondence with the set $V^\ell_a \cap H$ and so has the same size.

A graph refinement lemma
------------------------

We will need the following simple lemma, showing that any bipartite graph can be refined so that both vertex sets have high minimum degree (relative the to the original edge density).

\[lem-refine\] Let 