No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three

TL;DR

This paper proves that determining whether a Borel directed graph has a Borel dichromatic number at least three is a highly complex problem, with no countable basis, contrasting with simpler uncountable cases.

Contribution

It establishes the $oldsymbol ext{Pi}^1_2$-completeness of the Borel dichromatic number problem and shows no countable basis exists for graphs with finite Borel chromatic thresholds.

Findings

Deciding Borel dichromatic number ≥ 3 is $oldsymbol ext{Pi}^1_2$-complete.

No countable family of Borel directed graphs can serve as a basis for this class.

For every finite $k$, the set of Borel graphs with Borel $k$-coloring is $oldsymbol ext{Sigma}^1_2$-complete.

Abstract

I prove that the Borel directed graphs whose vertex set admits a partition into two Borel acyclic sets form a $\mathbf\Sigma^1_2$-complete set; equivalently, that deciding whether a Borel directed graph has Borel dichromatic number at least~$3$ is a $\mathbf\Pi^1_2$-complete problem. It follows that no countable family of Borel directed graphs can serve as a basis for this class under Borel homomorphism and, more generally, that any basis must be at least as complex as~$\mathbf\Pi^1_2$. The proof lifts the classical NP-completeness reduction of Bokal, Fijav\v{z}, Juvan, Kayll, and Mohar to the Borel setting, using the coding framework of Thornton. Combined with a straightforward reduction from undirected to directed coloring problems, this completes the picture for finite Borel chromatic and dichromatic thresholds: for every finite $k$, the set of Borel (directed) graphs admitting a Borel $k$-(di)coloring is $\mathbf\Sigma^1_2$-complete, and in particular admits no countable basis. This contrasts with the uncountable threshold, where a single-element basis exists for Borel chromatic number (Kechris--Solecki--Todor\v{c}evi\'c) and a continuum-size basis exists for Borel dichromatic number (Raghavan--Xiao).