Multipartite nearly orthogonal sets over finite fields

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, Milojevi\'{c} 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 $d \geq k \geq \ell + 1$, $\mathbb{F}^d$ contains a $(k,\ell)$-nearly orthogonal set of size $d^{\delta k / \log k}$. This nearly matches an upper bound $\binom{d+k}{k}$ 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 W\"otzel, and we provide a new and short proof for this lemma.