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

Let K be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space Rⁿ, there is a well-studied notion of "ultrametric orthogonality" in Kⁿ. In this paper, motivated by a question of Erd{{o}}s in the real case, given integers k ≥ ≥ 2, we investigate the maximum size of a subset S ⊆ Kⁿ \ 0\ satisfying the following property: for any E ⊆ S of size k, there exists F ⊆ E of size such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.