$H=(V_1\cup V_2 \cup V_3, E)$ is a complete $3$ partite graph such that $|V_1|=|V_2|=|V_3|=n$ . Color the edges with three colors.

My question is: Is it possible to find sets $V_1' \subset V_1, V_2' \subset V_2$ and $V_3' \subset V_3$, such that all edges between $V_1', V_2'$ and $V_3'$ have the same color and also $|V_i'|\geq \epsilon n$ for $i=1,2,3.$?

Could someone please give a reference for this problem?