27 problems

Let $n$ be even. In the rainbow perfect matching game $\mathcal{RP}_n$, played on $n/2$ copies of $K_n$, Maker wins by claiming a rainbow perfect matching; let $b_{\mathcal{RP}_n}$…

Let $C$ be a fixed graph containing three non-isolated vertices, and define its 2-density by … There are constants $c_1,c_2>0$ such that, for sufficiently large $n$, Maker wins the…

In a Maker–Breaker game and its corresponding Waiter–Client game, Maker and Waiter are the respective players seeking to claim a winning set. Beck's conjecture. Whenever Maker wins…

Let $C$ be a fixed graph. In the Maker–Breaker $C$-game on the edges of the complete graph $K_n$, Maker claims one edge per round and Breaker claims up to $q$ edges per round; Make…

Let $s\geq 3$ and let $\mathcal{D}_{s,n}$ be the $(1:b)$ Maker–Breaker game on the edges of $K_n$ in which Maker wins precisely when her spanning subgraph has diameter at most $s$.…

Mindegree-1 tightness conjecture. Maker has a randomized strategy to win with probability at least

Let $K_n$ be the complete graph on $n$ vertices. A graph is tree-universal at degree bound $D$ if it contains a copy of every tree $T$ with $n$ vertices and maximum degree…

A biased Maker–Breaker game is played by Maker and Breaker claiming elements of a board according to fixed rules; in a random-player version, the players act randomly, while in a p…

Let $k$ be a positive integer and let $p$ be an odd prime such that $k+1=p^2$. Let $f(k,-1)$ denote the least board size for which Maker can force a solution to the unit-fraction e…

Let $f^*(k,-1)$ be the least board size for which Maker can force a solution with distinct variables in the Maker–Breaker game for the equation $1/x_1+\cdots+1/x_k=1/y$, and let…

Rahman–Watson conjecture. Their results remain true for general 3-CNF formulas without the constraint that each clause has a spare variable, except that the required number of roun…

Let $H_{n,s,p}$ be the random $s$-uniform hypergraph obtained from the complete $s$-uniform hypergraph on $n$ vertices by retaining each edge independently with probability $p$. Co…

Local-first conjecture. Except for triangles and forests—and, for triangles, only in the unbiased game—the global reason for Maker's win appears no earlier than the local reason, w…

Exception conjecture. Forests and triangles as subgraphs of maximum 1-density are the only exceptions to the very strong connection between the vertex Maker–Breaker game and the ve…

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,p)$-crossing game be the Maker-Breaker game in which Maker and Breaker claim equal numbers of edges per roun…

Let $S_{m\times n}$ denote the $m\times n$ square grid, and let the $(p,q)$-crossing game be the Maker-Breaker game on this grid in which Maker claims $p$ edges for every $q$ edges…

Let $H$ be an $n$-vertex graph with bounded maximum degree, and consider the $(1:b)$ Maker-Breaker game on $E(K_n)$. The strategy of Theorem is conjectured to be optimal up to a co…

Let $K_n$ be the complete graph on $n$ vertices. In the $T$-tournament game on $K_n$, Maker and Breaker alternately claim edges, and Maker chooses an orientation for each edge she…

In a $(1:b)$ Maker–Breaker game, Maker and Breaker claim edges from the edge set of the complete graph $K_n$, and the critical bias $b^*$ is the maximal value of $b$ for which Make…

Let $K_n$ be the complete graph on $n$ vertices, and let ${\mathcal T}_n$ denote the family of spanning trees on these vertices. In the Maker–Breaker game $(E(K_n),{\mathcal T}_n)$…

Let $G$ be a $d$-regular graph on $n$ vertices, where $d\ge3$, and let $m$ be a positive integer. In the $(m:(d-2)m)$ game on $G$, Breaker’s conjecture. Breaker can force Maker to…

Stojaković–Szabó conjecture. There exists a constant $C$ such that for every

Chvátal–Erdős conjecture. The threshold bias is asymptotically

Stojaković–Szabó's conjecture. With high probability,

Let $n$ elements $\vec v_i$ of $\mathbb Z_{2n}^{d}$ be given for $i\in[n]$. A partition into pairs is a partition of $\mathbb Z_{2n}^{d}$ into elements $\vec x_i^j,\vec y_i^j$ inde…