# Hyperfiniteness and Borel combinatorics

@article{Conley2019HyperfinitenessAB,
title={Hyperfiniteness and Borel combinatorics},
author={Clinton T. Conley and Steve Jackson and Andrew S. Marks and Brandon Seward and Robin D. Tucker-Drob},
journal={Journal of the European Mathematical Society},
year={2019}
}
• Published 7 November 2016
• Mathematics
• Journal of the European Mathematical Society

### Banach-Tarski paradox using pieces with the property of Baire.

• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 1992
This note solves a problem of Marczewski from 1930 by showing that there is a partition of S2 into sets A1, Ak, B1,..., Bl with a different strong regularity property, the Property of Baire.

### Martin’s conjecture and strong ergodicity

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation

### Borel version of the Local Lemma

• Mathematics
• 2016
We prove a Borel version of the local lemma, i.e. we show that, under suitable assumptions, if the set of variables in the local lemma has a structure of a Borel space, then there exists a satisfying