Probabilistic threshold conjecture for non-2-colourability games

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices, and let $E(K_n^{(k)})$ be its edge set. Consider the $(1:q)$ Waiter–Client and Client–Waiter non-2-colourability games with winning set $\mathcal{NC}_2$. Denote their threshold biases by $b_{\mathcal{NC}_2}^{WC}$ and $b_{\mathcal{NC}_2}^{CW}$, respectively.

Non-2-colourability threshold conjecture. The threshold biases satisfy

$$\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{NC}_2}^{WC})^{-1}}{2^{k-1}\ln 2-\ln 2/2}\right\}=\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{NC}_2}^{CW})^{-1}}{2^{k-1}\ln 2-\ln 2/2}\right\}=1.$$

The conjecture predicts that both games have threshold biases asymptotically equivalent to the values suggested by the probabilistic intuition. The paper establishes bounds with an exponential gap in $k$ for the Waiter–Client version and a substantially sharper bound for the Client–Waiter version, leaving the stated asymptotics open.