Factor of iid Schreier decoration of transitive graphs

The paper proves that for each of the four even-degree Archimedean lattices (square, triangular, Kagome, (3,4,6,4)) there is a finitary Aut(Λ)-factor Schreier decoration with no infinite monochromatic paths, exactly as stated in Theorem 2 and carried out lattice-by-lattice; the construction ensures that each colour class is a union of finite cycles that are then strongly oriented . The hierarchy tools (Theorem 16 and Lemma 18) and the balanced-orientation result (Theorem 17) support the planar arguments and boundary management used in the constructions . The candidate solution reduces the task to producing a finitary Aut(Λ)-equivariant partition into d edge-disjoint 2-factors with finite cycles and then orients each cycle deterministically from the labels—this is precisely what the paper constructs and uses to conclude the Schreier decoration (e.g., the final “pick one of two strong orientations” step) , with analogous arguments for the triangular and Kagome/(3,4,6,4) cases . The only nit is a minor wording in the model equating a 2-factorization with a Schreier decoration before explicitly orienting; after the orientation step, they are indeed equivalent for this purpose.