A Concise Proof of the $L_0$ Dichotomy
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914442127605760 |
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| author | Matos-Wiederhold, Tonatiuh |
| author_facet | Matos-Wiederhold, Tonatiuh |
| contents | 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 $G_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Concise Proof of the $L_0$ Dichotomy Matos-Wiederhold, Tonatiuh Logic Combinatorics 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 $G_0$. |
| title | A Concise Proof of the $L_0$ Dichotomy |
| topic | Logic Combinatorics |
| url | https://arxiv.org/abs/2604.02589 |