A Missing lemmas for the proof of Theorem 3.1

The following proof is from Daniely and V ardi [15], and we give it here for completeness. By Lemma A.1, there exists a DNF formula We construct such an affine layer in Lemma A.2. At least one of the k size-n slices in z contains 0 more than once. We define the outputs of our affine layer as follows. Pr [z represents a hyperedge ] = n (n 1) ... (n k + 1) null 1 n null Pr null z Z null 1 2 log(n) .