Each 11-vertex graph without 4-cliques has a triangle-free 2-partition of vertices - Nedialkov, Evgeni; Nenov, Nedyalko (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1999)
Let $G$ be a graph, $cl(G)$ denotes the clique number of the graph $G$. By $G → (3,3)$ we denote that in any 2-partition $V_1 \cup V_2$ of the set $V(G)$ of his vertices either $V_1$ or $V_2$ contains 3-clique (triangle) of the graph $G; α = min{|V(G)|, G → (3,3) and cl(G) = 4}, β = min{|V(G)|, G → (3,3) and cl(G) = 3}$. In the current article, we consider graphs $G$ with the property $G → (3,3)$. As a consequence from proven results it follows that $α = 8 and β ≥ 12$.