A complexity problem for Borel graphs

TL;DR

The paper demonstrates the complexity of classifying Borel graphs with infinite Borel chromatic number, showing that their subgraph structure is highly complex and answering a question posed by Kechris and Marks.

Contribution

It proves that the set of closed subgraphs of the shift graph with finite Borel chromatic number is -complete, revealing the complexity of the classification problem.

Findings

No simple basis exists for Borel graphs with infinite Borel chromatic number.

The set of subgraphs with finite Borel chromatic number is -complete.

The result answers a question of Kechris and Marks.

Abstract

We show that there is no simple (e.g. finite or countable) basis for Borel graphs with infinite Borel chromatic number. In fact, it is proved that the closed subgraphs of the shift graph on $[\mathbb{N}]^{<\mathbb{N}}$ with finite (or, equivalently, $\leq 3$) Borel chromatic number form a $\mathbf{\Sigma}^1_2$-complete set. This answers a question of Kechris and Marks and strengthens several earlier results.