Complexity of Finite Borel Asymptotic Dimension

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbfΣ^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.

💡 Research Summary

The paper investigates the descriptive‑set‑theoretic complexity of the property “a locally finite Borel graph has finite Borel asymptotic dimension” (asdim B). The authors prove that this property is Σ¹₂‑complete, establishing maximal descriptive complexity within the projective hierarchy. The central technical contribution is a combinatorial characterization for graphs generated by a single Borel function f. Theorem 1.2 shows that for an acyclic Borel function f, the following are equivalent: (i) asdim B(G_f) is finite, (ii) asdim B(G_f)=1, and (iii) for every natural number r there exists a Borel r‑forward‑independent hitting set for f. An r‑forward‑independent hitting set H⊆X is a Borel set intersecting every forward orbit of f, with the additional requirement that any two points of H are separated by at least r steps along the orbit. This reduces the geometric question about asymptotic dimension to a purely combinatorial existence problem.

Using a recent Σ¹₂‑hardness transfer theorem (derived from Todorčević–Vidnyánszky and further refined by Färich‑Shinko‑Vidnyánszky), the authors construct, on the space of infinite subsets of ℕ, a Borel structure where the existence of such hitting sets is Σ¹₂‑complete. Lemma 3.4 establishes a tight correspondence between r‑forward‑independent hitting sets for f and Borel homomorphisms from the directed graph (X,→G_f) to a simple directed graph D_r on ℕ (edges either decrement by one or jump from 0 to any integer ≥ r). This translation allows the authors to apply the general completeness machinery and prove Theorem 3.1: the set of parameters σ for which the associated function f_σ admits Borel r‑forward‑independent hitting sets is Σ¹₂‑complete for each fixed r, and consequently the set of σ for which f_σ admits such sets for all r is also Σ¹₂‑complete. Combining this with Theorem 1.2 yields the main result (Theorem 1.1): the collection of locally finite Borel graphs with finite Borel asymptotic dimension is Σ¹₂‑complete.

The paper then applies these techniques to Borel constraint‑satisfaction problems (CSPs) for directed graphs generated by a single Borel function. For a finite digraph H (assumed sinkless), the set CSP_function_B(H) consists of codes of Borel functions f such that the directed graph →G_f admits a Borel homomorphism to H. Theorem 1.4 classifies the complexity of CSP_function_B(H) according to the structure of H: (i) if H contains a loop, CSP_function_B(H) is Π¹₁ (indeed every such →G_f maps to H); (ii) if H is ergodic (has a strongly connected component that is aperiodic) and loop‑free, CSP_function_B(H) is Σ¹₂‑complete; (iii) otherwise (non‑ergodic, loop‑free) the problem is again Π¹₁. This mirrors the known Borel CSP dichotomy for general Borel digraphs and connects to the LOCAL model of distributed computing: the existence of efficient LOCAL algorithms for LCL problems on oriented paths corresponds exactly to the Σ¹₂‑complete case, while the absence of such algorithms yields Π¹₁‑complete CSPs.

The paper concludes with several open questions, notably whether the Σ¹₂‑completeness of finite Borel asymptotic dimension extends to arbitrary Borel graphs (beyond those generated by a single function) and how the complexities of CSP_B(H) and CSP_function_B(H) relate in general. Overall, the work provides a deep link between measurable combinatorics, descriptive set theory, and computational complexity, introducing forward‑independent hitting sets as a powerful tool for analyzing Borel asymptotic dimension and Borel CSPs.