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## Ultraproducts of measure preserving actions and graph combinatorics

- Clinton T. Conley, A. Kechris, Robin D. Tucker-Drob
- MathematicsErgodic Theory and Dynamical Systems
- 16 February 2012

Abstract Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching… Expand

## Measurable chromatic and independence numbers for ergodic graphs and group actions

- Clinton T. Conley, A. Kechris
- Mathematics
- 31 January 2013

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… Expand

## A bound on measurable chromatic numbers of locally finite Borel graphs

- Clinton T. Conley, B. D. Miller
- Mathematics
- 2016

A graph on a set X is an irreflexive, symmetric set G ⊆ X ×X. Such a graph is locally finite if every point has only finitely many G-neighbors. A (κ-)coloring of such a graph is a function c : X → κ… Expand

## Borel asymptotic dimension and hyperfinite equivalence relations

- Clinton T. Conley, S. Jackson, Andrew S. Marks, Brandon Seward, Robin D. Tucker-Drob
- Mathematics
- 14 September 2020

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In… Expand

## BROOKS’ THEOREM FOR MEASURABLE COLORINGS

- Clinton T. Conley, Andrew S. Marks, Robin D. Tucker-Drob
- MathematicsForum of Mathematics, Sigma
- 13 January 2016

We generalize Brooks’ theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d\geqslant 3$ which contains no $(d+1)$ -cliques, then $G$ admits a ${\it\mu}$… Expand

## Følner tilings for actions of amenable groups

- Clinton T. Conley, S. Jackson, David Kerr, Andrew S. Marks, Brandon Seward, Robin D. Tucker-Drob
- Mathematics
- 3 April 2017

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G (“shapes”) with prescribed… Expand

## Unfriendly colorings of graphs with finite average degree

- Clinton T. Conley, O. Tamuz
- MathematicsProceedings of the London Mathematical Society
- 12 March 2019

In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree… Expand

## Equitable Colorings of Borel Graphs

- Anton Bernshteyn, Clinton T. Conley
- Mathematics
- 27 August 2019

Hajnal and Szemeredi proved that if $G$ is a finite graph with maximum degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a proper coloring with $k$ colors in which every two… Expand

## Measure-theoretic unfriendly colorings

- Clinton T. Conley
- Mathematics
- 2014

We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the… Expand

## MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

- Clinton T. Conley, Andrew S. Marks, Spencer Unger
- MathematicsForum of Mathematics, Sigma
- 12 March 2019

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include… Expand

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