Abstract
Abstract We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property and show that the analogous result for digraphs fails.
Highlights
One may consider the Borel chromatic number of, or ( ), defined as the least cardinal that admits a standard Borel structure with respect to which there is a Borel -coloring of . (Note that a standard Borel structure exists on iff ∈ {0, 1, 2, . . . , א0, 2א0 }, and for each such, it is unique up to Borel isomorphism.)
One of their primary successes was the isolation of a Borel graph G0 on 2N of uncountable Borel chromatic number that admits a continuous homomorphism to every analytic Borel graph on a Polish space of uncountable Borel chromatic number
Suppose that is an analytic graph on a Polish space
Summary
We construct a basis for Borel digraphs with Borel chromatic number > 2. Let ⊆ be an analytic set, and assume that there exists an ∈ N such that if , ∈ and is a -walk of odd length from to , dilength( ) ≤. -invariant Borel set ( , ) ⊇ [ ( , )] and a Borel 2-coloring , of ↾ ( , ) Assume that such an does not exist; we will show that is not -terminal. Using the fact that Σ( ) = dilength( ) > ( ) · ≤ |Σ( ( ))|, one can deduce that there exists a unique one-step extension ′ of that is compatible with This contradicts the assumption that was -terminal. It is easy to check that is a Borel map defined on , while the -invariance of the sets ( , ) implies that is a 2-coloring. ◦ (( , ( ))
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