Erdős Problem #1175 — Triangle-Free Subgraphs of Prescribed Chromatic Number

Let $\kappa$ be an uncountable cardinal, i.e. $\aleph_0<\kappa$. Must there exist a cardinal $\mu$ such that every graph $G$ with chromatic cardinal $\mu$ contains a subgraph $H$ that is triangle-free and has chromatic cardinal exactly $\kappa$? Equivalently, is the following true: for every cardinal $\kappa$ with $\aleph_0<\kappa$, there exists a cardinal $\mu$ such that, for every vertex type $V$ and every graph $G$ on $V$ with chromatic cardinal $\mu$, there is a subgraph $H$ of $G$ such that $H$ contains no triangle and its chromatic cardinal is $\kappa$?