A Concise Proof of the $L_0$ Dichotomy

TL;DR

The paper presents a shorter, graph-theoretic proof of the $L_0$ dichotomy, replacing transfinite analysis with a Borel reflection argument and a $\sigma$-ideal framework.

Contribution

It introduces a simplified proof of the $L_0$ dichotomy using a $\sigma$-ideal approach and Borel reflection, improving upon the original transfinite method.

Findings

A new proof of the $L_0$ dichotomy is established.

The proof employs a $\sigma$-ideal of small sets of homomorphisms.

Largeness preservation is shown via the First Reflection Theorem.

Abstract

Carroy, Miller, Schrittesser, and Vidny\'anszky established the $L_0$ dichotomy: there is a Borel graph of Borel chromatic number three that admits a continuous homomorphism to every analytic graph of Borel chromatic number at least three. Their proof relies on a transfinite analysis of terminal approximations over a decreasing $\omega_1$-sequence of analytic sets. I give a new, substantially shorter proof of this result by adapting the graph-theoretic framework recently introduced by Bernshteyn for the $G_0$ dichotomy. The central device is a $\sigma$-ideal of \emph{small} sets of homomorphisms from finite path approximations into the target graph, where smallness is witnessed by a bounded odd-walk condition on vertex projections. The key lemma that largeness is preserved under the doubling operation is established via the First Reflection Theorem, replacing the original transfinite construction with a single Borel reflection argument. The continuous homomorphism from the canonical graph $\mathbb{L}_c$ into the target is then obtained as a limit of shrinking families of copies, in direct analogy with Bernshteyn's proof for $G_0$.