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 Erdős in the real case, given integers \(k \ge \ell \ge 2\), we investigate the maximum size of a subset \(S \subseteq {\mathcal {K}}^n {\setminus }\{\textbf{0}\}\) satisfying the following property: for any \(E \subseteq S\) of size k, there exists \(F \subseteq E\) of size \(\ell \) such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.