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

Abstract Let $${\mathcal {K}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $${\mathbb {R}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , there is a well-studied notion of “ultrametric orthogonality” in $${\mathcal {K}}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . In this paper, motivated by a question of Erdős in the real case, given integers $$k \ge \ell \ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , we investigate the maximum size of a subset $$S \subseteq {\mathcal {K}}^n {\setminus }\{\textbf{0}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>\</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> satisfying the following property: for any $$E \subseteq S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> of size k , there exists $$F \subseteq E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> of size $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.