Past exam of the model theory course of the University of Oxford / 2011 / 2 / a / iii / Solution

A model-complete theory is a first-order theory for which every structure embedding between two of its first-order models is an elementary embedding. Thus, if $f:\mathcal{A}\hookrightarrow \mathcal{B}$ is such an embedding, then for every first-order formula $\phi (\mathbf{x})$ and tuple $\mathbf{a}$ from $A$,first-order model completeness does not assert that every two first-order models are elementarily equivalent: it constrains embeddings that exist between first-order models, whereas completeness constrains all first-order models.