The acyclic Client graph conjecture for the Waiter–Client game

Let $K_n$ be the complete graph on $n$ vertices. In a $(q:1)$ Waiter–Client game on $E(K_n)$, Waiter offers $q+1$ unclaimed edges in each round, and Client claims one offered edge while Waiter claims the others. Acyclic Client graph conjecture. For any constant $\varepsilon>0$ and integer $q\geq(1+\varepsilon)n$, Client has a strategy to keep his graph acyclic.

This conjecture asserts that the proved sufficient condition $q\geq1.1n$ can be extended to every fixed margin above $n$. The paper proves that Waiter can force Client to build a cycle when $q\leq(1-\varepsilon)n$, while Client's strategy is established only for the larger constant $1.1$; the asymptotically tight threshold remains open.