19 problems

Let $X$ be a separable Banach space, let $A\subseteq X$ be Borel, and let $\mathbb{A}_{\mathsf{G}}$ be the auxiliary game referred to in the paper, with $\mathbb{A}$ denoting the c…

Let $X$ be a separable Banach space over $\mathbb{R}$, let $A\subseteq X$ be Borel, let $\mathcal{A}$ denote the ideal of Aronszajn-null sets, and let $\mathcal{A}_{\mathsf{G}}$ be…

Let $X$ be a separable Banach space over $cmathbb{R}$, let $A\toolbox X$ be Borel and Aronszajn-null, and let $\mathcal{A}_{\mathsf{G}}$ be the game in which player $\mathbf{I}$ ch…

Let $\mathbb B$ be a complete Boolean algebra, and let $\mathcal G^\textsf{\textup{c\&c}}_{<\aleph_{\omega+1}}(\mathbb B)$ denote the cut-and-choose game of length less than…

Strict hierarchy conjecture.

Let $T\subseteq \mathbb{Y}$ be a subtree, and write … For $d>\frac{1}{2}$ and $\delta\in\left[\frac{1}{2},d\right]$, consider win-lose alternating-move games on $T$ with payoff set…

Let ${\mathsf{AD}}$ denote the axiom of determinacy and let ${\mathsf{Bl}\text{-}\mathsf{AD}}$ denote the axiom of Blackwell determinacy. Martin's conjecture.…

Judge-winning characterization conjecture. If the judge has a winning strategy in the game $G_2(\kappa,\lambda)$, then $\kappa$ is $\lambda$-ineffable in Abe's sense.

In the Banach–Mazur game on a topological space, players successively choose nested nonempty open sets, and the second player wins when the resulting intersection lies in a target…

ETOG representation conjecture. Every bi-positional prefix-independent winning condition coincides on periodic sequences with some energy condition over a totally ordered group.

Kopczyński's conjecture. Prefix-independent half-positional winning conditions are closed under finite unions.

For an integer $n>1$, consider the $n$-for-$n$ variation of infinite Hex, in which Red and Blue each place $n$ stones at every turn, starting from an empty board. Equal-move conjec…

In infinite chess, a position has a game value given by the ordinal rank measuring how many moves a player can force before winning, when such a value is defined. Evans–Hamkins con…

Consider infinite Hex on the triangular lattice, with Red and Blue claiming unoccupied board positions according to the usual rules, but with a board position containing finitely m…

Horizontal initial-position conjecture. A game of Infinite Hex with initial position as in figure is a win for Red.

Red's one-tile advantage conjecture. A game of Infinite Hex with exactly one tile of the board assigned to Red at the initial position, and none to Blue, is a win for Red.

Let $\lambda\geq 2$ be a cardinal and let $A\subseteq{}^{\omega}\lambda$. In the multiboard Gale–Stewart game … the players play the ordinary Gale–Stewart game on $\kappa$ boards s…

Let $X$ be a topological space, let $\pi(X)$ be the least cardinality of a $\pi$-base for $X$, and let $\operatorname{BM}(X)$ denote the Banach–Mazur game on $X$. For positive card…

Let $A$ be the action set and let $(P_n)_{n\geq 0}$ be information partitions satisfying perfect recall, with Blackwell-game information: at even stages $n=2k$, player 1 observes t…