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.

令\({\mathcal {K}}\)为具有有限留数域的离散值域。与欧几里得空间\({\mathbb {R}}^n\)中的正交性类比， \({\mathcal {K}}^n\)中存在经过充分研究的“超度量正交性”概念。在本文中，受实际情况中 Erdős 问题的启发，给定整数\(k \ge \ell \ge 2\) ，我们研究子集\(S \subseteq {\mathcal {K}} ^n {\setminus }\{\textbf{0}\}\)满足以下性质：对于任意大小为k 的\(E \subseteq S\) ，存在大小为\( \ell \)使得F中的任何两个不同向量都是正交的。还研究了该特性的其他变体。