A determinacy approach to Borel combinatorics

We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable...

We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ t...

We study finite automata running over infinite binary trees. A run of such an automaton is usually said to be accepting if all its branches are accepting. In this article, we relax the notion of accepting run by allowing a certain quantity of rejecting branches. More precisely we study the following criteria for a run to be accepting: - it contains at most finitely (resp countably) many rejecting branches; - it contains infinitely (resp uncountably) many accepting branches;...

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $\Delta>2$ it is impo...

We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Borel versions of greedy algorithms and augmenting procedures, ...

We show, assuming weak large cardinals, that in the context of games played in a proper class of moves, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of $L$ that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seve...

We characterize countable dimensionality and strong countable dimensionality by means of an infinite game.

Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that the class of planar infinite matroids does not have a universal matroid. The existence of these matroids has a connection to Set Theory in that it corresponds to the Determinacy of certain games. To show that our construction gives matroid...

This paper is a continuation of the study of topological properties of omega context free languages (omega-CFL). We proved before that the class of omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there exist some omega-CFL which are analytic but non Borel sets. We prove here that there exist some omega context free languages which are Borel sets of infinite (but not finite) rank, giving additional answer to questions of Lescow and Thomas [Logical Speci...

We study infinite two-player win/lose games $(A,B,W)$ where $A,B$ are finite and $W \subseteq (A \times B)^\omega$. At each round Player 1 and Player 2 concurrently choose one action in $A$ and $B$, respectively. Player 1 wins iff the generated sequence is in $W$. Each history $h \in (A \times B)^*$ induces a game $(A,B,W_h)$ with $W_h := \{\rho \in (A \times B)^\omega \mid h \rho \in W\}$. We show the following: if $W$ is in $\Delta^0_2$ (for the usual topology), if the incl...