18 problems

Let $G$ be the evolving random graph process, and write $\tau(\mathcal E)$ for the hitting time at which event $\mathcal E$ first occurs. Let ${\cal M}$ be the perfect matching gam…

Let ${\cal H}$ be the set of Hamiltonian cycles in $K_n$. For a family of winning sets ${\cal F}$, let $b_{\cal F}^p$ denote the smallest bias for which Breaker wins the random gam…

Let $G$ be a random graph, and let $\gamma_{MB}(G,b)$ and $s_{MB}(G,b)$ denote the two Maker–Breaker domination game parameters. For a Binomial random graph $G_{n,p}$ with constant…

Beck's conjecture. For sufficiently large $n$ and every integer $t\geq 3$, there is a constant $c_t$, depending only on $t$, such that

Consider impartial Tak played on an $n\times n$ board, with the first player seeking a winning strategy. Odd-board first-player conjecture. If $n>5$ is odd, then the first player h…

Let $L(n)$ denote the quantity measuring the size of a minimal snake diagram. Quadratic lower-bound conjecture. For all $n$, … for some constants $a$ and $b$. This conjecture asser…

Let $n\geq 6$. Write $M_n$ for the value given by Proposition, and let $x$ satisfy … A penult is a position on an $n\times n$ board with the stated token count from which the relev…

Let $K_n$ be the complete graph on $n$ vertices. In the $(1:1)$ Client–Waiter game, Waiter offers two board elements at each round and Client claims one; Client's claimed edges for…

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…

Odd-cycle conjecture. For every odd integer $n\ge5$,

Let $H$ be a graph with $e(H)\ge 1$, and let $m(H)$ denote its maximum density over nonempty subgraphs, namely … Consider the $b$-Waiter–Client game on the edges of $K_n$, where Wa…

Let $b_\Delta$ denote the threshold bias for the biased Maker–Breaker $H$-factor game, where $H$ has maximum degree $\Delta$. Threshold conjecture. For all $\Delta \ge 3$, … This p…

Let $\mathcal{OC}_n$ denote the family of odd cycles on the board of the Client-Waiter game, and let $b_{cw}(\mathcal{OC}_n)$ be its threshold bias. Hefetz, Krivelevich and Tan's c…

Let $G$ be the target graph from the surrounding discussion, and let $\bigsqcup_\omega K_n$ denote a countable disjoint union of copies of the complete graph $K_n$. In the strong R…

Let $H$ be a graph with at least three edges. For an edge $e$ of $H$, let $H-e$ be the graph obtained by deleting $e$, together with a vertex of degree one if that vertex is an end…

Let $H$ be a unicyclic graph, meaning a connected graph containing exactly one cycle, and let $v(H)$ denote its number of vertices. Let $f^+_H(n)$ be the upper threshold of the str…

For a non-empty graph $H$, define … where $v(F)$ and $e(F)$ are the numbers of vertices and edges of $F$, respectively. Let $f^-_H(n)$ be the lower threshold of the strict…

Let $G$ be a hypergraph. Suppose that Breaker wins the Maker-Breaker game on $G$, with Maker moving first. Beck's Chooser-Picker conjecture. Picker wins the Chooser-Picker game on…