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

TL;DR

This paper investigates the maximum size of subsets in discrete valued fields with ultrametric orthogonality properties, extending concepts from Euclidean orthogonality and addressing a question inspired by Erdős.

Contribution

It introduces bounds on the size of ultrametric orthogonal sets over discrete valued fields, exploring various orthogonality conditions and their combinatorial limits.

Findings

Established upper bounds for the size of ultrametric orthogonal sets.

Analyzed variants of the orthogonality property and their implications.

Extended Euclidean orthogonality concepts to discrete valued fields.

Abstract

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 Erd{\H{o}}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ 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.