Approximate Schreier decorations and approximate Kőnig’s line coloring Theorem

Following recent result of L. M. Tóth [arXiv:1906.03137] we show that every 2∆-regular Borel graph G with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate König’s line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs. It is a standard fact from finite combinatorics that every 2∆-regular finite graph is a Schreier graph of the free group F∆ on ∆ generators. This means that every such graph admits an orientation and a ∆-labeling of the edges such that for every α ∈ ∆ and every vertex there is exactly one out-edge with label α and exactly one in-edge with label α. Such an orientation and labeling is called a Schreier decoration. Note that every Schreier decoration corresponds to an action of the free group F∆ on the vertex set of the graph. We refer the reader to the introduction in [11] for more information about Schreier decorations. The analogous statement for infinite graphs without any restriction on definability follows from the axiom of choice. In the measurable setting, i.e., when the vertex set is endowed with a standard probability (Borel) structure and we require the orientation and labeling to be measurable, the full analogue of the statement fails. This follows from the example of Laczkovich [9] who constructed an acyclic 2-regular bipartite graph on the unit interval that is not induced by an action of Z on any set of a full measure. However, Tóth recently proved [11] that if the measure is invariant one can always find a measurable Schreier decoration on a different graph that has the same local statistics. This can be stated in a compact form as follows: every 2∆-regular unimodular random rooted graph has an invariant random Schreier decoration, see [11, Theorem 1]. An equivalent formulation in a language that is closer to the one in this paper is as follows, see [11, Corollary 4]: Every 2∆-regular graphing (G, μ) is a local isomorphic copy of some graphing (G ′, μ′) that is induced by a Borel action of F∆ that preserves μ ′. The key steps in the proof of [11, Theorem 1] are (I) a consequence of [11, Theorem 3]: for every ∆-regular bipartite graphing (G, μ) and for every > 0 there is a Borel map c : E → ∆ that is a proper edge coloring on a set of μ-measure at least 1− , The author was supported by Leverhulme Research Project Grant RPG-2018-424.