In this paper we apply the method of ultraproducts to the study of graph combinatorics associated with measure preserving actions of infinite, countable groups, continuing the work in Conley-Kechrisâ€¦ (More)

A countable group Î“ is called shift-minimal if every non-trivial measure preserving action of Î“ weakly contained in the Bernoulli shift Î“ y ([0, 1] , Î» ) is free. We show that any group Î“ whoseâ€¦ (More)

We show that for any infinite countable group G and for any free Borel action G y X there exists an equivariant class-bijective Borel map from X to the free part Free(2G) of the 2-shift G y 2G. Thisâ€¦ (More)

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so thatâ€¦ (More)

This note answers a question of Kechris: if H < G is a normal subgroup of a countable group G, H has property MD and G/H is amenable and residually finite then G also has property MD. Under the sameâ€¦ (More)

We study actions of countable discrete groups which are amenable in the sense that there exists a mean on X which is invariant under the action of G. Assuming that G is nonamenable, we obtainâ€¦ (More)

We generalize Brooksâ€™s theorem to show that if G is a Borel graph on a standard Borel space X of degree bounded by d â‰¥ 3 which contains no (d + 1)-cliques, then G admits a Î¼-measurable d-coloringâ€¦ (More)

A measure preserving action of a countably infinite group Î“ is called totally ergodic if every infinite subgroup of Î“ acts ergodically. For example, all mixing and mildly mixing actions are totallyâ€¦ (More)