I am trying to understand the proof of Theorem 2 given here. (Page 5) The theorem states that $\forall k\exists$ a triangle free graph $G$ with $\chi(G)>k$. The proof constructs such a $G$ as $G=A_kA_{k-1}\cdots A_0(G_0)$, where $A_0(G_0)$ is defined as the amalgamation of the graph $G_0$(which is a previously constructed triangle free $(k+1)$-partite graph having partite sets $V_0,\cdots V_k$, with the property that if it is $k$-colored with each partite set monochromatic then there is a monochromatic edge) on $V_0$.
What is not clear to me is why $G$ will contain a copy of $G_0$ with each class monochromatic. By a previous proposition $A_0(G_0)$ contains a copy of $G_0$ with $V_0$ monochromatic, say red. Now $A_1A_0(G_0)$ contains a copy of $A_0(G_0)$ with its second partite set monochromatic say blue. This does not mean that $V_1$ of $G_0$ is blue.
I would be grateful if someone could resolve my confusion. Thanks for reading through this post.