# Publications

**Lacunary sets and actions of tsi groups**. Submitted.

**Recurrence and the existence of invariant measures**. With Manuel Inselmann. Submitted.

**Edge sliding and transverse hyperfinite decompositions**. With Anush Tserunyan. Submitted.

**On the existence of cocycle-invariant Borel probability measures**. Submitted.

**Bases for functions beyond the first Baire class**. With Raphaël Carroy. Submitted.

**Open graphs, Hurewicz-style dichotomies, and the Jayne-Rogers theorem**. With Raphaël Carroy. Submitted.

**Incomparable actions of free groups**. With Clinton Conley. To appear in *Ergodic theory and dynamical systems*.

**Sigma-continuity with closed witnesses**. With Raphaël Carroy. *Fundamenta Mathematicae*, 29-42, **239**, 2017.

**Measurable perfect matchings for acyclic locally countable Borel graphs**. With Clinton Conley. *Journal of Symbolic Logic*, 258-271, **82 (1)**, 2017.

**Dichotomy theorems for families of non-cofinal essential complexity**. With John Clemens and Dominique Lecomte. *Advances in Mathematics*, 285-299, **304**, 2017.

**Orthogonal measures and strong ergodicity**. With Clinton Conley. *Israel Journal of Mathematics*, 83-89, **218 (1)**, 2017.

**Measure reducibility of countable Borel equivalence relations**. With Clinton Conley. *Annals of Mathematics*, 347-402, **185 (2)**, 2017.

**A bound on measurable chromatic numbers of locally finite Borel graphs**. With Clinton Conley. *Mathematical Research Letters*, 1633-1644, **23 (6)**, 2016.

**The smooth ideal**. With Clinton Conley and John Clemens. *Proceedings of the London Mathematical Society*, 57-80, **112 (1)**, 2016.

**An embedding theorem of
E**_{0} with model theoretic applications. With Itay Kaplan. *Journal of Mathematical Logic*, 1450010, **14 (2)**, 2014.

**The Borel cardinality of Lascar strong types**. With Itay Kaplan and Pierre Simon. *Journal of the London
Mathematical Society*, 609-630, **90 (2)**, 2014.

**Essential countability of treeable equivalence relations**. With John Clemens and Dominique Lecomte. *Advances in Mathematics*, 1-31, **265**, 2014.

**An antibasis result for graphs of infinite Borel chromatic number**. With Clinton Conley. *Proceedings of the American Mathematical Society*, 2123-2133, **142 (6)**, 2014.

**Stationary probability measures and topological realizations**. With Clinton Conley and Alexander Kechris. *Israel Journal of Mathematics*, 333-345, **198 (1)**, 2013.

**Incomparable treeable equivalence relations**. *Journal of Mathematical Logic*, 121-162, **12 (1)**, 2012.

**The graph-theoretic approach to descriptive set theory**. *Bulletin of Symbolic Logic*, 554-575, **18 (4)**, 2012.

**Definability of small puncture sets**. With Andres Caicedo, John Clemens, and Clinton Conley. *Fundamenta Mathematicae*, 39-51, **215 (1)**, 2011.

**Dichotomy theorems for countably infinite dimensional analytic hypergraphs**. *Annals of Pure and Applied Logic*, 561-565, **162 (7)**, 2011.

**A classical proof of the Kanovei-Zapletal canonization**. *Contemporary Mathematics*, 281-285, **533**, American Mathematical Society, 2011.

**Descriptive Kakutani equivalence**.
With Christian Rosendal.
*Journal of the European Mathematical Society*, 179-219, **12 (1)**, 2010.

**Ends of graphed equivalence relations, II**.
With Greg Hjorth.
*Israel Journal of Mathematics*, 393-415, **169**, 2009.

**Ends of graphed equivalence relations, I**. *Israel Journal of Mathematics*, 375-392, **169**, 2009.

**Basis theorems for non-potentially closed sets and graphs of uncountable Borel chromatic number**.
With Dominique Lecomte.
*Journal of Mathematical Logic*, 121-162, **8 (2)**, 2008.

**Measurable chromatic numbers**. *Journal of Symbolic Logic*, 1139-1157, **73 (4)**, 2008.

**The existence of quasi-invariant measures of a given cocycle, II: Probability measures**. *Ergodic Theory and Dynamical Systems*, 1615-1633, **28 (5)**, 2008.

**The existence of quasi-invariant measures of a given cocycle, I: Atomless, ergodic sigma-finite measures**. *Ergodic Theory and Dynamical Systems*, 1599-1613, **28 (5)**, 2008.

**Means on equivalence relations**. With Alexander Kechris. *Israel Journal of Mathematics*, 241-263, **163**, 2008.

**Coordinatewise decomposition of group-valued Borel functions**. *Fundamenta Mathematicae*, 119-126, **196 (2)**, 2007.

**Isomorphism of Borel full groups**.
With Christian Rosendal.
*Proceedings of the American Mathematical Society*, 517-522, **135 (2)**, 2007.

**Coordinatewise decomposition, Borel cohomology, and invariant measures**. *Fundamenta Mathematicae*, 81-94, **191 (1)**, 2006.

**Topics in orbit equivalence**.
With Alexander Kechris. *Lecture Notes in Mathematics*, **1852**, Springer, 2004.