BOREL FRACTIONAL COLORINGS OF SCHREIER GRAPHS

The paper’s main theorem (Theorem 1.1) states exactly the two claims the candidate proves: for any countable group Γ, finite F, and free Borel action Γ ↷ X, one has χ*_{B}(G(X,F)) ≤ 1/α_β(G(Free(2^Γ),F)), and, in particular, equality holds when X = Free(2^Γ) . The paper gives a direct proof of the inequality via a clopen-approximation lemma (Lemma 2.1) and a coding/averaging construction using a Borel function f on DD^{-1}-neighborhoods , together with the general lower bound χ*_{B}(G) ≥ 1/α_μ(G) for any probability measure μ on V(G) . The model’s solution, by contrast, reduces (1) to the Bernoulli free part using monotonicity under Γ-equivariant Borel maps and the Seward–Tucker-Drob class-bijective map (a deduction approach that the paper explicitly notes is possible from (2) ), and then invokes the equality on the Bernoulli free part. Substantively, both are correct and consistent; they differ chiefly in method: the paper proves the upper bound directly, while the model deduces it from the equality case via the Seward–Tucker-Drob mapping. Minor issues: (i) in the model’s explicit finite-Γ embedding, the displayed formula for Φ should use g_x^{-1} to be equivariant with the left shift; (ii) the model’s brief proof sketch for the Bernoulli equality mentions LP duality/LLL, which is not how this paper’s short proof proceeds. These do not affect the outcome. Overall, the statements align and are correct.