A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such induced subgraph $H$ of $\omega(H) 2$ satisfies $\chi(H) 4$. This disproves well-known conjecture. Our construction digraph bounded clique number, large dichromatic no directed cycles odd length at least 5.