For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove that for every prime $p$ there exists some $\delta = \delta(p)>0$, such that for every field $\mathbb{F}$ of characteristic $p$ and for all integers $k \geq 2$ and $d \geq k$, there exists a $k$-nearly orthogonal set of at least $d^{\delta \cdot k/\log k}$ vectors of $\mathbb{F}^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set ${\cal A}$ of non-self-orthogonal vectors of $\mathbb{F}^d$ such that for every two subsets of ${\cal A}$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set ${\cal A}$ include $\ell+1$ pairwise orthogonal vectors for an arbitrary fixed integer $1 \leq \ell \leq k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.

翻译：对于一个域 $\mathbb{F}$ 和整数 $d$ 与 $k$，若集合 ${\cal A} \subseteq \mathbb{F}^d$ 中的向量均非自正交，且其中任意 $k+1$ 个向量都包含一个正交对，则称该集合为 $k$-近正交的。我们证明：对于每个素数 $p$，均存在某个 $\delta = \delta(p)>0$，使得对于任意特征为 $p$ 的域 $\mathbb{F}$ 以及所有满足 $k \geq 2$ 和 $d \geq k$ 的整数，在 $\mathbb{F}^d$ 中均存在一个 $k$-近正交集合，其包含至少 $d^{\delta \cdot k/\log k}$ 个向量。该集合的规模在指数项的 $\log k$ 因子内是最优的。我们进一步证明了此结果的两种推广形式。其一，我们构造了 $\mathbb{F}^d$ 中一个庞大的非自正交向量集 ${\cal A}$，使得对于任意两个大小为 $k+1$ 的子集，其中一个子集的某个向量与另一个子集的某个向量正交。其二，对于任意固定的整数 $1 \leq \ell \leq k$，所构造集合 ${\cal A}$ 中任意 $k+1$ 个向量都包含 $\ell+1$ 个两两正交的向量。证明过程涉及概率与谱方法以及超图容器方法。