Finding the largest 3-clique-free induced subgraph
Both definitions leave your problem NP-hard, and have been answered on cstheory.


Interpretation 1, where you require the largest triangle free sub-graph, is NP-Hard and has been answered here.
Intepretation 2, where you need a partition such that both the induced sub-graphs are triangle free, has been answered here.


Note that the answers I linked to are for general $H$-freeness and are a class of $NP$-Hard problems.