Measurable chromatic and independence numbers for ergodic graphs and group actions

Abstract

We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan's property (T), and freeness. We also prove a Borel analog of the classical Brooks' Theorem in finite combinatorics for actions of groups with finitely many ends.

Ultraproducts of measure preserving actions and graph combinatorics

“…(i) When is amenable, this follows from the result of Abért et al [ACLT] that Cay( , S) admits a perfect matching, using also the quasi-tiling machinery of Ornstein and Weiss [OW], as in Conley and Kechris [CK,4.10,4.11]. The second case follows immediately from Rokhlin's lemma.…”

“…Indeed, this fails for = Z or = (Z/2Z) * (Z/2Z) (with the usual set of generators S for which d = 2) and a the shift action of on 2 , since then the shift action of on Col(2, , S) with this random coloring would be mixing and then as in (i) ⇒ (ii) of Proposition 7.2, by taking b to be also mixing, one could have a mixing action a ∈ FR( , X, µ) for which there is a measurable 2-coloring, which easily gives a contradiction. On the other hand, it follows from the result in [CK,5.12] that was mentioned earlier, that for any with finitely many ends, except for = Z or = (Z/2Z) * (Z/2Z), one indeed has for any a ∈ FR( , X, µ) an invariant, random d-coloring which is a factor of the action a. We do not know if this holds for groups with infinitely many ends.…”

“…This is easily false for some action a (e.g. the shift action), when = Z or = (Z/2Z) * (Z/2Z) (with the usual sets of generators) and it was shown in Conley and Kechris [CK,5.12] that when has finitely many ends and is not isomorphic to Z or (Z/2Z) * (Z/2Z), then one indeed has the Brooks bound χ µ (S, a) ≤ d for any a ∈ FR( , X, µ) (in fact, even for Borel as opposed to measurable colorings). It is unknown if this still holds for with infinitely many ends, but Theorem 6.2 shows that one has the full analog of the Brooks bound up to weak equivalence for any group .…”

“…A compactness argument using Brooks' theorem also shows that χ (S, a) ≤ d, where χ (S, a) is the chromatic number of G(S, a). It was shown in Conley and Kechris [CK,2.19,2.20] that for any infinite , χ ap µ (S, a) ≤ d for any a ∈ FR( , X, µ), so one has a full 'approximate' version of Brooks' theorem. How about the full measurable Brooks bound χ µ (S, a) ≤ d?…”

“…By Lemma 7.9, it is a Borel graph whose connected components are isomorphic to Cayley graphs of degree d = |S ±1 | that have finitely many ends. So, by Conley and Kechris [CK,5.1,5.7,5.11] and Lemma 7.9, (X R , E) has a Borel d-coloring C R : X R → {1, . .…”