"Gap 1" two-cardinal principles and the omitting types theorem for ℒ(Q)

For λ a strong limit singular cardinal, and more generally for λ > 2^{cof λ}, we prove the equivalence of a number of model theoretic and combinatorial conditions, including the ℒ(Q)-completeness theorem for the λ ⁺-interpretation, an omitting types theorem for ℒ(Q) in the λ ⁺-interpretation, and a weak form of Jensen's principle □_λ.