We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.

Abstract. 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 approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz-HuczekZhang tiling theorem for countable amenable groups and strengthens the Ornstein-Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations.In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.

We generalize Brooks' theorem to show that if G is a Borel graph on a standard Borel space X of degree bounded by d 3 which contains no (d + 1)-cliques, then G admits a µ-measurable dcoloring with respect to any Borel probability measure µ on X , and a Baire measurable d-coloring with respect to any compatible Polish topology on X . The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID d-colorings of Cayley graphs of degree d, except in two exceptional cases.

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k ≥ 1 and A, B ⊆ R k are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of Z d .