For a field \(\mathbb {F}\) and integers d, k and \(\ell \) , a set \(A \subseteq \mathbb {F}^d\) is called \((k,\ell )\) -nearly orthogonal if all vectors in A are non-self-orthogonal and every \(k+1\) vectors in A contain \(\ell + 1\) pairwise orthogonal vectors. Recently, Haviv, Mattheus, Milojević and Wigderson have improved the lower bound on nearly orthogonal sets over finite fields, using counting arguments and a hypergraph container lemma. They showed that for every prime p and an integer \(\ell \) , there is a constant \(\delta (p,\ell )\) such that for every field \(\mathbb {F}\) of characteristic p and for all integers \(d \ge k \ge \ell + 1\) , \(\mathbb {F}^d\) contains a \((k,\ell )\) -nearly orthogonal set of size \(d^{\delta k / \log k}\) . This nearly matches an upper bound \(\left( {\begin{array}{c}d+k\\ k\end{array}}\right) \) coming from Ramsey theory. Moreover, they proved the same lower bound for the size of a largest set A where for any two subsets of A of size \(k+1\) each, there is a vector in one of the subsets orthogonal to a vector in the other one. We prove a common generalisation of this result, showing that essentially the same lower bound holds for the size of a largest set \(A \subseteq \mathbb {F}^d\) with the stronger property that given any family of subsets \(A_1, \ldots , A_{\ell +1} \subseteq A\) , each of size \(k+1\) , we can find a vector in each \(A_i\) such that they are all pairwise orthogonal. Rather than combining both counting and container arguments, we make use of a multipartite asymmetric container lemma that allows for non-uniform co-degree conditions. This lemma was first discovered by Campos, Coulson, Serra and Wötzel, and we provide a new and short proof for this lemma.
