A determinacy approach to Borel combinatorics

We start by giving a survey to the theory of Borel*(\kappa) sets in the generalized Baire space Baire({\kappa}) = {\kappa}^{\kappa}. In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel({\kappa}), \Delta^1_1({\kappa}) and {\Sigma}^1_1({\kappa}) and the connections between Borel*(\kappa)-sets and the infinitely deep language M_{{\kappa}^+{\kappa}}. In the end of the paper we prove the consistency of Borel*(\kappa) ...

n infinite two-player zero-sum game with a Borel winning set, in which the opponent's actions are monitored eventually but not necessarily immediately after they are played, is determined. The proof relies on a representation of the game as a stochastic game with perfect information, in which Chance operates as a delegate for the players and performs the randomizations for them, and on Martin's Theorem about the determinacy of such games.

In 1998 Zielonka simplified the proofs of memoryless determinacy of infinite parity games. In 2018 Haddad simplified some proofs of memoryless determinacy of finite parity games. This article adapts Haddad's technique for infinite parity games. Two proofs are given, a shorter one and a more constructive one. None of them uses Zielonka's traps and attractors.

The game tree languages can be viewed as an automata-theoretic counterpart of parity games on graphs. They witness the strictness of the index hierarchy of alternating tree automata, as well as the fixed-point hierarchy over binary trees. We consider a game tree language of the first non-trivial level, where Eve can force that 0 repeats from some moment on, and its dual, where Adam can force that 1 repeats from some moment on. Both these sets (which amount to one up to an obv...

This paper has been merged with arXiv:0908.3765

In Problem #1542 of Mathematics Magazine, Grossman and Turett define the Cantor game. In his 2007 Mathematics Magazine article about the Cantor game, Matt Baker proves several results and poses three challenging questions about it: Do there exist uncountable subsets of [0, 1] for which: Alice does not have a winning strategy; Bob has a winning strategy; neither Alice nor Bob has a winning strategy? In this paper we show that the answers to these questions depend upon whic...

We explore a version of the minimax theorem for two-person win-lose games with infinitely many pure strategies. In the countable case, we give a combinatorial condition on the game which implies the minimax property. In the general case, we prove that a game satisfies the minimax property along with all its subgames if and only if none of its subgames is isomorphic to the "larger number game." This generalizes a recent theorem of Hanneke, Livni and Moran. We also propose seve...

In this work, I investigate several combinatorial properties of Borel (or co-analytic) ideals on countable sets. I extend a theorem presented in M. Hru\v{s}\'{a}k, D. Meza-Alc\'antara, E. Th\"ummel, and C. Uzc\'ategui, \emph{Ramsey Type Properties of Ideals}, and also find an $F_\sigma$ tall ideal in which player II has a winning strategy in the Cut and Choose Game, which was a question posed by J. Zapletal. In the second section, I present some Ramsey properties of ideals, s...

We prove a game theoretic dichotomy for $G_{\delta\sigma}$ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis' proof of $G_{\delta\sigma}$ determinacy.

This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so called "Baire category" and the "$\sigma $-category". This is mainly done by means of (generalized) Banach-Mazur games using the Axiom of determinacy (the inconsistency of AC and AD is reflected in the beginning and a weaker form of AC is chos...