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Is there a two-coloring of the vertices of F which does not contain a monochromatic G as a subgraph? show that this problem is contained

Is there a two-coloring of the vertices of F which does not contain a monochromatic G as a subgraph?

show that this problem is contained in one of the classes in the second level of the polynomial hierarchy

P.S:The previous answer given for this question is incorrect.Please provide a correct and detailed explaination

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