Triangle-free graphs with arbitrarily large chromatic number ($\chi (M_k)=k$)

Define $M_2=K_2$ and $M_{k+1}=\mu (M_k)$ for $k\ge 2$. By the Mycielskian chromatic increase and Mycielskian triangle preservation, each $M_k$ is a finite triangle-free graph with chromatic number $k$. The first steps are $K_2,C_5$, the Grötzsch graph, and then a 23-vertex graph needing five colours. The vertex recurrence $n_{k+1}=2n_k+1$ gives $n_k=3\cdot 2^{k-2}-1$. These explicit examples show how arbitrarily strong graph coloring constraints can accumulate without any three-vertex clique.