The paper introduces a new type of examples of bounded degree acyclic Borel graphs, called homomorphism graphs, and studies their combinatorial properties in the context of descriptive combinatorics. The main result shows that for ∆> 2, the family of ∆-regular acyclic Borel graphs with Borel chromatic number at most ∆forms a Σ1
2-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.
The key idea is to associate an acyclic Borel graph Homac(T∆, H) to a given Borel graph H, where the vertex set consists of pairs (x, h) with x a vertex in H and h a homomorphism from the ∆-regular infinite rooted tree T∆to H that maps the root to x. The authors show that the Borel chromatic number of Homac(T∆, H) is controlled by the weakly provably ∆1
2-chromatic number of H.
Using this construction, the authors establish several applications:
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