Bounded width characterizes the easy side of Borel homomorphism problems

Establish that for every finite relational structure H, the Borel homomorphism problem CSP_B(H) is Π^1_1 if and only if it is not Σ^1_2-complete if and only if H has bounded width in the classical CSP sense (i.e., all instances are solvable by local consistency/datalog algorithms).

Background

The paper proves that, contrary to the classical finite CSP dichotomy, solving systems of linear equations over a finite field is already Σ^1_2-complete in the Borel setting. Combined with results of Thornton, this shows that homomorphism problems that are not of bounded width are necessarily Σ^1_2-complete in the Borel context.

Motivated by this separation, the authors propose that bounded width may exactly delineate the boundary between the “easy” (Π^1_1) and “hard” (Σ^1_2-complete) Borel homomorphism problems, mirroring the role of polymorphisms and width in the algebraic theory of classical CSPs.