In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure \leq rather than \subseteq. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic or rich model. The specifics of \leq determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.

Three conjectures

The initial applications of the Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have:

The construction

Let L be a finite relational language. Fix C a class of finite L-structures which are closed under isomorphisms and substructures. We want to strengthen the notion of substructure; let \leq be a relation on pairs from C satisfying: Definition. An embedding is strong if Definition. The pair has the amalgamation property if then there is a so that each B_i embeds strongly into D with the same image for A. Definition. For infinite D and we say A \leq D iff A \leq X for Definition. For any the closure of A in D, denoted by is the smallest superset of A satisfying Definition. A countable structure G is -generic if: Theorem. If has the amalgamation property, then there is a unique -generic. The existence proof proceeds in imitation of the existence proof for Fraïssé limits. The uniqueness proof comes from an easy back and forth argument.