A determinacy approach to Borel combinatorics

This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.

First proved my Donald Martin in 1975, the result of Borel determinacy has been the subject of multiple revised proofs. Following Martin's book [1], we present a recent streamlined proof which implements ideas of Martin, Moschovakis, and Hurkens. We aim to give a concise presentation that makes this proof approachable to a wider audience.

We prove a number of results on the determinacy of $\sigma$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $\sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}\omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $\sigma$-projective sets from opt...

We propose a new determinacy hypothesis for transfinite games, use the hypothesis to extend the perfect set theorem, prove relationships between various determinacy hypotheses, expose inconsistent versions of determinacy, and provide a philosophical justification for determinacy.

The study of inner models was initiated by G\"odel's analysis of the constructible universe. Later, the study of canonical inner models with large cardinals, e.g., measurable cardinals, strong cardinals or Woodin cardinals, was pioneered by Jensen, Mitchell, Steel, and others. Around the same time, the study of infinite two-player games was driven forward by Martin's proof of analytic determinacy from a measurable cardinal, Borel determinacy from ZFC, and Martin and Steel's p...

We study infinite two-player games where one of the players is unsure about the set of moves available to the other player. In particular, the set of moves of the other player is a strict superset of what she assumes it to be. We explore what happens to sets in various levels of the Borel hierarchy under such a situation. We show that the sets at every alternate level of the hierarchy jump to the next higher level.

It is shown that Borel games of length $\omega^2$ are determined if, and only if, for every countable ordinal $\alpha$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $\alpha$-many iterated powersets above a limit of Woodin cardinals.

This is an introduction into John Conway's beautiful Combinatorial Game Theory, providing precise statements and detailed proofs for the fundamental parts of his theory. (1) Combinatorial game theory, (2) the GROUP of games, (3) the FIELD of numbers, (4) ordinal numbers, (5) games and numbers, (6) infinitesimal games, (7) impartial games.

In a game where both contestants have perfect information, there is a strict limit on how perfect that information can be. By contrast, when one player is deprived of all information, the limit on the other player's information disappears, admitting a hierarchy of levels of lopsided perfection of information. We turn toward the question of when the player with super-perfect information has a winning strategy, and we exactly answer this question for a specific family of lopsid...

We characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorics or structural properties of the given filter. These generalize several ultrafilter games of Galvin.