6 problems

Let $B_{2,n}$ denote the two-spine book graph with $n$ pages, and let $\operatorname{HGP}(G)$ be the proper hat-guessing number of a graph $G$. Two-spine book graph conjecture. For…

Let $\widehat{\mathcal{S}}_1$ be the class of standard-imaginary strategies and let $\widehat{\mathcal{M}}_1$ be the class of imaginary strategies used in the continuous Levine gam…

Let $V_{2,h}$ be the optimal winning probability for the two-player Levine hats game with finite hat-stack height $h$. Exponential convergence conjecture. There exist constants…

Let $V_n$ denote the optimal winning probability in the initial Levine hats game with $n$ players. The sequence $(V_n)_{n\geq 1}$ is non-increasing and has a limit in $[0,1/2]$. Le…

Let $G$ be a cycle $C_n$ and let $h$ be a hat function such that $2 \leq h(v) \leq 4$ for every vertex $v$. Let $A\in V(G)$ be such that $h(A)=2$. The game…

Let $\Gam$ be a directed visibility graph with vertex set $V(\Gam)=V$, and let $V_v$ be the nonempty set of colours available to vertex $v$. Write $\deg_-(v)$ for the indegree of…