13 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 $\mathcal{RS}_n$ be the rainbow-spanning-tree game played on $n-1$ copies of $K_n$, where Maker wins by claiming a rainbow spanning tree, and let $b_{\mathcal{RS}_n}$ denote it…

For the rainbow-connectivity game $\mathcal{C}_{s,n}$, let $b_{\mathcal{C}_{s,n}}$ denote its threshold bias. Assume that $s=\omega(\log(n))$. Rainbow-connectivity threshold conjec…

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

Strict monotonicity conjecture. For fixed $b,m$, for all sufficiently large $N$,

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…

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 $K_n$ 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