The infinite critical tournaments conjecture

A tournament $T$ is $k$-clique-critical if $\operatorname{\overrightarrow{\omega}}(T)=k$ and $\operatorname{\overrightarrow{\omega}}(T-v)=k-1$ for every $v\in V(T)$. The infinite critical tournaments conjecture. For every integer $k\geq 3$, there is an infinite number of $k$-clique-critical tournaments. The paper explains that this conjecture is tied to the stronger cluster question: if true it would refute that question, while if false it would support it; the conjecture itself remains open.