Client–Waiter tree-universality conjecture

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 form her graph. A graph is tree-universal at degree bound $D$ if it contains a copy of every tree $T$ with $n$ vertices and maximum degree $\Delta(T)\leq D$. Client–Waiter tree-universality conjecture. There exists a constant $c>0$ such that, for every sufficiently large integer $n$, in the $(1:1)$ Client–Waiter game on $K_n$, Client has a strategy to build a graph that is tree-universal at degree bound $\frac{cn}{\log(n)}$. The source notes that current arguments yield weaker degree bounds and that the Waiter–Client proof does not presently adapt to this setting; the conjecture remains open in the supplied text.