On the maximum size of ultrametric orthogonal sets over discrete valued fields

Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space ${\mathbb{R}}^n$, there is a well-studied notion of “ultrametric orthogonality” in ${\mathcal{K}}^n$. In this paper, motivated by a question of Erd{ő}s in the real case, given integers $k\ge \mathrm{\ell }\ge 2$, we investigate the maximum size of a subset $S\subseteq {\mathcal{K}}^n\setminus \{\mathbf{0}\}$ satisfying the following property: for any $E\subseteq S$ of size $k$, there exists $F\subseteq E$ of size $\mathrm{\ell }$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied.