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A Concise Proof of the $L_0$ Dichotomy

Tonatiuh Matos-Wiederhold

Abstract

Carroy, Miller, Schrittesser, and Vidnyánszky 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 $ω_1$-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 $σ$-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 $G_0$.

A Concise Proof of the $L_0$ Dichotomy

Abstract

Carroy, Miller, Schrittesser, and Vidnyánszky established the 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 -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 dichotomy. The central device is a -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 into the target is then obtained as a limit of shrinking families of copies, in direct analogy with Bernshteyn's proof for .

Paper Structure

This paper contains 5 sections, 9 theorems, 10 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Finite path approximations
  3. A notion of smallness for copies
  4. Construction of a minimal graph of Borel chromatic number at least 3
  5. Proof of Theorem \ref{['thm:L0-hom-equivalent-main']}

Key Result

Theorem 1.2

If $G$ is a Borel graph on a Polish space all of whose degrees are bounded by the natural number $k$, then $\chi_B(G)\leq k+1$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: $L_3^c$ for $c(0)=1,c(1)=3,c(2)=5$.

Theorems & Definitions (16)

  • Theorem 1.2: Proposition 4.6 in kechris1999borel
  • Definition 1.3
  • Lemma 1.4: First Reflection Theorem
  • Definition 3.1
  • Lemma 3.2: Claim 3.5 of L0paper
  • Theorem 3.3
  • proof : Proof of theorem
  • Lemma 3.4
  • proof
  • Claim 3.4.1
  • ...and 6 more