Optimization problems and the polynomial hierarchy

It is demonstrated that such problems as the symmetric Travelling Salesman Problem, Chromatic Number Problem, Maximal Clique Problem and a Knapsack Packing Problem are in the Δ^P₂ level of PH and no lower if ∑^P₁ ≠ Π^P₁, or NP≠co-NP. This shows that these problems cannot be solved by polynomial reductions that use only positive information from an NP oracle, if NP≠co-NP. It is then shown how to extend these results to prove that interesting problems are properly in Δ^P,X_i+1 for all X,k where ∑^P,X_k≠Π^P,X_k in PH^X.