Borel distinguishing number

TL;DR

This paper explores the Borel distinguishing number of Borel graphs, revealing differences from the classical distinguishing number and providing examples that demonstrate these distinctions at various levels.

Contribution

It introduces the concept of the Borel distinguishing number and constructs examples showing its divergence from the classical distinguishing number.

Findings

Existence of Borel graphs with countable distinguishing number but uncountable Borel distinguishing number.

For every integer n ≥ 3, there exists a Borel graph with distinguishing number 2 and Borel distinguishing number at least n.

Abstract

In this paper, we study definable variants of the notion of the distinguishing number of a graph in descriptive set theoretic setting. We introduce the notion of the Borel distinguishing number of a Borel graph and provide examples that separate distinguishing number and Borel distinguishing number at various levels. More specifically, we prove that there exist Borel graphs with countable distinguishing number but uncountable Borel distinguishing number and that, for every integer $n \geq 3$, there exists a Borel graph with distinguishing number $2$ whose Borel distinguishing number is finite and at least $n$.