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

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